QA 

.U8 



















4-' "k 




AN INTRODUCTIO]^ 



TO 



GEOMETET 



UPON THE 



ANALYTICAL PLAN. 



V»^ 


BY 

F. H. LOUD, 


\ 




Professor in Colorado College. 


i 
--4 



ST. LOUIS: 
G. I. JONES AND COMPANY, 

18 80. 




w\ 



Entered according to Act of Congress, in the year 1880, by 

F. H. LOUD, 
In the Office of the Librarian of Congress, at Washington. 



I 



*^/<^ 



PREFACE. 



In -the preparation of the present text-book for the use of 
beginners in Geometry, the needs of two different classes of 
students have been held in mind. 

The primary design of the book, in its inception, was to furnisk 
an Introduction to a text-book in Analytical Geometry, by the 
same author, already in great part prepared, and soon to be 
printed, provided the present book meet with a sufficiently 
favorable reception. It is thus intended that students in the 
higher schools — those aiming at an extended mathematical 
course, whether in college or elsewhere — shall find in this httle 
book the opportunity to render that course homogeneous from the 
beginning^ and at the same time greatly to dbhreviate its initial 
stages. It frequentty happens that a student, after painfully 
acquiring the method of Euchd only to find that it is to be laid 
aside for that of Descartes, experiences so much difficulty in the 
transition as never to become truly familiar with Algebraic Geom- 
etry or to appreciate the beauty of its processes. By using the 
first fifty-two pages of this text-book as an introduction to Ana- 
lytical Geometry, the student finds his geometrical course, from 
its most elementary definition to the remotest application of the 
Calculus, only the orderly and harmonious development of a single 
subject according to a uniform plan. 

The second class of students for whom it is hoped that this 
book may supply a need comprises those who aim at a practical 
use of elementary geometrical principles, and require only so 
much of the theory as will suffice to render this use intelligent. 
Such students will find the essential problems of land surveying 
and similar applications of Geometry brought in this text-book as 
near the threshold of the subject as their nature will admit. 

As an Introduction to Analytical Geometry the first one hundred 

(ill) 



IV PREFACE. 

and eleven sections are considered sufficient. The subsequent sec- 
tions are appended with the view of rendering the present work 
complete in itself by embracing all the topics usually treated in 
Elementary Geometries and Plane Trigonometries, — and thia 
largely for the use of the second class of students mentioned. In 
the author's Analytical Geometry those sections of the present 
book which follow Sect. Ill will either be incorporated in their 
proper places, or replaced by other methods of demonstration. 
The latter will be the course pursued, for instance, in regard to 
the Circle and all volumes bounded by curved surfaces ; where 
the methods of Analytical Geometry proper, combined with the 
principle of the Center of Gravity, will furnish simpler demon^ 
strations than could be brought within the compass of the present, 
work. 

Having thus defined the persons for whom and the objects for 
which this Introduction is especially intended, the author desires 
to say to any who may adopt it one urgent word in regard to the 
manner of its use. The ancient method of Euclid has unques- 
tionably a great merit in this, that the actual forms dealt with are 
kept continually in the eye and mind of the student by its constant 
dependence upon diagrams. The diagrams employed in the dem- 
onstrations of this text-book are few, proofs from formulae being 
always preferred for their generality. It is of the highest import- 
ance, therefore, that in every case of an algebraic demonstration 

THE STUDENT ILLUSTRATE AND TEST ITS APPLICATION BY DIAGRAMS- 
DRAWN BY HIMSELF iu as great a variety of form as possible. 
Unless this principle be applied throughout, the use of this book 
will entail failure from the lack of clear and intuitive ideas of the 
subjects presented, but by faithfully pursuing this course it is- 
believed that all the advantages of the old and new methods may 
be combined. 

It has been thought unwise to encumber this book with trigo- 
nometrical tables further than is necessary for illustration. When 
required for practical use, it is recommended to employ, in con- 
nection with the text of this Introduction, the volumes of tables 
by Prof. Loomis or Prof. Pierce. 

F. H. L. 
Colorado Springs, Sept. 1, 1880. 



CONTENTS. 



PAGB 

Primary Definitions 1 

Angles 5 

Systems of Lines ^ 

Functions of Angles 12 

Determination of Points in a Plane 1& 

Polar Coordinates 20 

Cartesian Coordinates 28 

Signs of the Functions of Angles 26 

Functions of the Sum and Difference of Angles, and of 

Multiple Angles 32 

Particular Values of the Functions 38 

Table of Natural Sines and Tangents 45-48 

Mean Ordinates 52 

Area 54 

Polygons 58 

Plane Trigonometry 63- 

The Circle 71 

Miscellaneous Examples 79 

Geometry of Three Dimensions 80 

Mensuration of Volumes 87 



In the notation of angles in this work the letters of the Greek 
alphabet are employed, which are as follows : — 



a Alpha. 

ft Beta. 

Y Gamma, 

d Delta, 

e Epsilon. 

C Zeta. 

7) Eta. 

6 Theta. 



( 


Iota. 


X 


Kappa. 


X 


Lambda. 


P- 


Mu. 


V 


Nu. 


^ 


Xi. 





Omicron, 


TZ 


Pi. 



p Rho. 

c Sigma 

T Tau. 

i> Upsilon. 

<p Phi. 

X Chi. 

(p Psi. 

Hi Omega^ 



(Yii> 



Mathematics is the science of measurement. 

Geometry is the mathematics of space. 

Geometry, therefore, is founded on the axioms of quantity 
which are common to all mathematics, and with which the student 
has become f amihar in Algebra, and on certain additional concep- 
tions either involved in the idea of space, or inseparably associated 
with it. They resemble the proper axioms, or necessary truths, 
in this respect, that the mind accepts argument based upon them 
without questioning their validity, and usually without being 
aware that an admission has been made. Among these principles 
two may be instanced as of special importance : — 

No point can he at once in two different directions from another 
point. 

Any number of points may move in any direction without change 
of their relative directions or distances. 



<viii' 



GEOMETRY. 



PRIMARY DEFINITIONS. 

1. K point is that which has no size, but only position. 

Show that a dot merely represents a point. 

2. If a point changes its position, or moves, its path is 
called a line. 

Illustrated by the ordinary way of *' drawing lines." The end of the pencil 
represents the moving point. Show that there is no breadth. 

3. The moving point is called the describing or generat- 
ing point, and the line is said to be described bj it. 

4. Any one of the successive positions occupied by the 
generating point is a point on the line. 

5. A limited portion of a line is included between two 
points, and is called a distance. (See § 10.) 

6. Lines are of three kinds, distinguished as follows : 
(1.) When the generating point moves constantly in the 

same direction, the line it describes is called a straight line, 
or a right line. 

(2.) AVhen the generating point constantly changes its 
direction, the line described is called a curved line, or a 
curve, 

(3.) When the generating point changes its direction 
only at intervals, the line described is called a broken line. 
(See Fig. 1.) 

Illustrated by the manner in which each is drawn. 

7. A broken line obviousl}^ is made up of several parts, each of 
which is a portion of a right line. 

(1) 



2 PRIMARY DEFINITIONS. 

A curve may be considered as a broken line, in which these 
parts are extremely short. 

8. Any right line may be considered as described in 
either of two opposite directions. 

The generating point may be supposed to move in one direction, 
to some indefinite extent, then to return on the same path, when 
it will, of course, describe the same line in an opposite direction. 

Hence, a right line may be considered as described from any 
point on it, in both directions. 

9. A right line must always be understood to be unlim- 
ited in extent. 

Hence, we cannot speak of measuring a right line, but only dis- 
tances upon it. A point is often designated by a letter of the 
alphabet, — then a right Hne is designated by naming any two- 
points upon it, in the order in which the line is considered as. 
described. The line BA, Fig. 1, is the same line as AB, but 
the directions indicated are opposite. A distance is also denoted 
by naming the points at each extremity ; but, to avoid confusion, 
a vinculum will be used when this is intended. Thus, AB de- 
notes a line passing through A and B ; AB denotes the distance 
between these points. 

10. Distance is thus far spoken of as a concrete quantity, — a 
portion of a line. The term is also used to denote an abstract 
number, — viz., the number of times any portion of a line con- 
tains that portion, or distance, which is used as a unit of measure- 
ment. In the latter sense, a distance may be denoted b}^ any 
arithmetical or algebraic symbol for a quantity, and is subject 
(as concrete numbers are not) to every arithmetical or algebraic 
process, such as raising to powers, extracting roots, etc. The 
sense must be determined by the connection. 

11. As a moving point describes a line, so a moving line- 
generates a surface. 

Any point of a line may be considered to move along some 
other line, carrying the first line witli it. The place through 
which the line moves is then called a surface, the moving line is 



PEIMARY DEFINITIONS. 6 

called the generating line, or the generatrix, and the other line is 
called the directrix. The various kinds of surfaces depend on 
the form of these two lines, — whether they be straight, broken, 
or curved. 

Illustrated by the use of carpenter's tools, etc. Show that there is length 
and breadth only. 

12. When both the generatrix and directrix are right 
lines, and the former retains always the same direction, the 
surface generated is called a plane. 

As the lines are unlimited (§ 9) so is the plane. 

13. When a plane is generated, any point of the generat- 
ing line describes a right line having the same direction as 
the directrix. 

See Fig. 2, where AB is the path along which the point O of 
the generating line DC moves. Any other point on DC, as G, 
must move in the same direction as O, since DC is fixed in direc- 
tion. 

14. It thus appears that through any point of a plane may be 
drawn two right lines which shall be wholly in the plane, one 
having the same direction as the directrix, and the other having 
the direction of the generatrix ; for the latter may be regarded as. 
one of the successive positions which the generatrix assumes. 
But lines lying wholly within the plane may be drawn, not merely 
in these two directions, but in an indefinite number of different 
directions. In Fig. 2, suppose the plane to be described by the 
generatrix DC, whose point O moves along the directrix AB. 
Suppose that when the moving line has reached the position IK 
an obstacle intervenes at the point H, upon which the generatrix is 
forced to turn as on a pivot, the point O moving as before along 
AB. The motion of the generatrix no longer conforms to the- 
requirement of § 12, but it will be seen that no point of the gen- 
eratrix can reach any position which would not have been reached! 
had the motion continued unchecked ; hence all the varying posi- 
tions of the generatrix lie wholly in one plane. Moreover, if any 
two points of a plane be given, one of them may be assumed as a 



4 PKIMARY DEFINITIONS. 

pivot upon which the generatrix may be conceived to swing, as 
above described, until it passes through the other point ; hence, 

A straight line joining two points of a plane lies vv^hollj 
in the plane. 

15. If, after the generatrix has swung to the position FL, the 
obstacle at H be removed, and the motion continued without sub- 
sequent change of the direction of the generatrix, then the same 
plane will be generated as if the original motion had continued 
without any obstacle. For since both generatrix and directrix 
are of unlimited extent, there is no point which would be reached 
by the generatrix in the one position which would not be reached 
in the other. Hence, 

Any line of a plane may be taken as a generatrix. 

Similarly, 

Any line of a plane may be taken as a directrix. 

But the directrix and generatrix must not extend in the same 
direction. 

16. As a portion of a line is included between points, so 
a portion of a surface must be included within lines, or por- 
tions of them. When the including lines have such forms 
or positions that the included portion of the surface is finite 
in extent, it is called ^figure, 

17. As a moving line generates a surface, so any limited 
portion of a plane may move along a line not lying in that 
plane ; and that which is thus produced is called a volume. 

An entire plane thus moving along an unlimited line must pass 
through the whole of space. Hence a volume is a limited portion 
of space. As a figure is included within parts of lines, so a volume 
is included within parts of surfaces, which may be that by which 
it was generated (in two different positions), and those generated 
by the line or lines including the generating figure. 

N. B. — In the following §§ whenever lines are men- 
tioned, right lines are to be understood, unless the contrary 



ANGLES, 



<] 



is stated ; whatever is said of distances is to be understood 
of distances on right lines ; and whatever is said of lines, 
points, etc., is to be understood of lines, points, etc., in one 
plane, 

ANGLES. 

18. An aoigle is the difference in direction of two lines, 
or the amount of divergence between two lines that pass 
through one point. (Fig. 3.) 

To obtain the true idea of the magnitude of an angle, conceive 
two straight lines, of unlimited length, proceeding from a common 
point ; let one of them be fixed in direction, and let the other be 
turned about the common point. When the two lines coincide, 
their angle is 0, and it increases as they separate. In the same 
way the addition of angles is illustrated, for this increase is a pro- 
cess of addition. 

19. When two or more lines have the same direction, the 
angle between them is zero, and they are said to he parallel. 
(See Fig. 2, EP and AB.) 

20. If, from a given point, two lines proceed in exactly 
opposite directions, their divergence is the greatest possible, 
and is taken as the standard for the measurement of angles. 

An angle half as great as this is called a right angle. 

Illustrated by the points of the compass. 

The ninetieth part of a right angle is called a degree [°], 
the sixtieth part of a degree is called a minute ['], and the 
sixtieth part of a minute is called a second ["]. 

Hence, when a right line is considered as described (§8) from 
any point in it, in opposite directions, the angle between the two 
parts of the line on opposite sides of this point contains two right 
angles, or 180°. 

Angles containing less than 90° are called acute, those 
containing more than 90°, obtuse, angles. 

21. When the sum of two angles is equal to 180°, they 
are said tq be supplements of each other ; when their sum is 
equal to 90°, they are called complements of each other. 



\) ANGLES. 

22. When two lines intersect (see Fig. 3), the four 
angles formed are named as follows : — 

Those on the same side of one line, but on opposite sides 
of the other, are called adjacent angles, as a and /?. 

Those on opposite sides of both lines are called opposite^ 
or vertical^ angles, as a and a', or /? and /5'. 

23. To investigate their relations, consider each line as de- 
scribed in both directions from the point of intersection, A. 
Then the sum of the angles a and /? is evidently the angle between 
AE and AC, which is 180°. (See § 20.) Hence, 

Adjacent angles are supplements of each other. 

24. Both /S and /?' are adjacent to a, hence each is the supple- 
ment of a ; . •. a + /5' = a 4~ /^? o^ /^' = /^- That is, 

Vertical angles are equal to each other. 

25. li a — 90°, then 180° — a = 90°, or, a right angle is equal 
to its supplement. When one of the angles at the intersection of 
two lines is a right angle, the other three are also right angles, 
since two of them are supplements to it, and the third is vertical 
to it. 

When the angles between two lines are right angles, the 
lines are said to be perpe7idicular to each other. 

26. Since an angle is the difference in direction of two 
lines, any number of lines having the same direction will 
make equal angles with any line that crosses them. 

Thus, in Fig. 4, if AB, CD, etc., are parallel, they will all 
make the same angles with IJ ; and if KL is parallel to IJ, the 
angles it makes with AB, CD, etc., are equal to those IJ makes 
with them, a, a", etc., each equals a, and /?, /5', /5", etc., each 
equals 180° — a. The same principle employed in §§ 23, 24, 
determines which angles are equal and which are supplements. 
The rule may be stated thus (compare § 22) : 

Those angles which lie on corresponding sides ot both 
the lines which form them, or on opposite sides of both, are 



ANGLES. 7 

equal ; those which lie on corresponding sides of one, and 
opposite sides of the other, are supplements. 

Thus, in Fig. 4, a and d and d\ which all lie to the riglit of one 
and above the other of the lines which form them, are equal ; d" 
lies to the left of one, and helow the other Une forming it, hence 
this is equal to either of the former three; but d and /5' each lie 
to the riglit of one of their lines, while one is above and the other 
below the other line, hence these two angles are supplements. 

27. When three lines intersect at different points, the 
angles between any two of them are distinguished as 
interior and exterior angles. 

The interior angle is that produced when both the lines 
forming it are described from their intersection toiuard the 
third line. 

The exterior angle is an adjacent angle to the interior, 
and therefoi'e (§23) always its supplement. 

Thus, in Fig. 5, when BA is described toward the line FE, 
and CD also toward that line, the difference between their direc- 
tions is a which is the interior angle betweeen the lines. (5, or d\ 
which is produced when one of the same lines is described from 
their intersection toward FE and the other away from it, is the 
exterior angle, a', which is equal to the interior angle (§ 24), 
has no specific name. 

28. When three lines intersect at different points, the 
exterior angle which two of them make with each other is 
equal to the sum of the interior angles which they make with 
the third line. 

In Fig. 6, the exterior angle C adjacent to a is equal to the sum 
of the interior angles /5 and y. For through the intersection A let 
a hne pass parallel to PQ, and dividing the angle 'C into two parts 
d and e. Now, d = ,5 and r = ^ (§ 26), hence C = <1 + e z=: /5 + ^. 

29. When three lines intersect at different points, the 
sum of the interior angles is equal to 180°. 

For a + : im 180° (§ 23), but : == /S + /- (§ 28) . •. a + /5 + ^ =: 
180°. 



8 ANGLES. 

30. If two of the lines are perpendicular to each other, the 
angle between them is equal to 90°, hence the sum of the other 
two angles must be equal to ( 180° — 90°) or 90°. That is : — 

When a line intersects two other lines that are perpen- 
dicular to each other, the interior angles which it makes 
with them are complements. 

31. Two lines perpendicular to the same line are parallel. 

In Fig. 7 let AB and CD each be perpendicular to BD, then 
are they parallel. For, if hot, suppose some other line, as ED, 
passing through D, to be parallel to AB ; then (§ 26) the angle 
which it makes with BD is equal to a, that is, to 90°. But, by 
supposition, the angle made by CD with BD is 90°, hence these 
two angles are equal, i. e., the part to the whole, an absurdity; 
therefore CD is parallel to AB. 

Hence, only one perpendicular can be drawn from a point 
to a line. 

For if two lines, passing through the same point, have the same 
direction, obviously they coincide in one line. 

32. It is of special importance that the pupil give deflniteness to his own 
conceptions by drawing figures to illustrate the principles learned, as far as 
possible. For this purpose he needs at this stage of his progress no other 
instruments than he can make for himself, in accordance with the foregoing 
principles. 

Let a piece of tolerably stiff paper be folded once ; the edge of the fold will 
represent a straight line. Now fold it again, taking care that the edge of the 
former fold is turned accurately upon itself. The angle where the two folds 
meet will be a right angle. Por the two angles on each side of the second fold 
are equal, since the folding has made them coincide, and together they make 
up the divergence of the two parts of a straight line. (^ 20.) 

With this instrument (which we will call a square) the following problems 
may be solved : — 

A. To draw a line perpendicular to a given line, from a given point upon it. 
Place the square at the given point, so that one edge falls on the given line, the 
other edge will serve as a ruler by which to draw the perpendicular. 

B. To draw a line perpendicular to a given line, from a given point without 
it. Place the square so that one edge coincides with the given line, and slide 
it upon the latter, until the other edge passes through the given -point. 

C. To draw a line parallel to a given line, through a given point. Draw any 
line perpendicular to the given line, then through the given point draw a line 
perpendicular to this. (§ 31.) 



SYSTEMS OF LI^ES. 9 

By means of these rules, let the pupil construct illustrative figures for all the 
following demonstrations, varying the form as much as possible from that given 
on the chart. 

For some subsequent uses, it will be convenient to have a scale of equal 
parts drawn on each edge of the square. These should be copied from some 
standard, as a foot-rule, so that all may be alike. A fourth or a fifth of an inch 
is a convenient unit, if the drawing is to be on paper. The right angle of the 
square should be marked 0, and the equal divisions counted in each direction 
from this point. 

SYSTEMS OF LINES. 

33. A system of lines consists of any number of lines 
having some mutual relation. Thus, if a number of lines 
all have the same direction, they are called a system of 
parallels ; if they all pass through one point they are called 
a system of converging lines, or of convergents, and the 
common point is called the point of convergence. 

34. If two or more lines, forming a system of converg- 
ents, are intersected by another line, that part of any one 
of the convergents included bet\veen the intersecting line 
and the point of convergence is called the intercejpt of the 
intersecting line on that convergent. 

Fig. 8 represents a system of three convergents, AB, AC and 
AD. AG- is the intercept of MN on AD, AF is its intercept on 
AC, etc. 

35. The parts of two parallels, included between two 
other parallels, are equal. 

Take a line (EF, Fig. 9,) of one of the systems as the gener- 
ating line of the plane in which both are situated (§ 15), and a 
line, AB, of the other system for the directrix. The line CD 
will then be described by the point J (§ 13). Now, GH has the 
same direction with EF ; therefore, if EF moves to the right 
(retaining its direction), when the point I reaches the point K the 
two lines will coincide, and the point J will fall on the point L. 
Therefore the distances IJ and KL must be equal. 

36. Let any number of points be taken at equal distances 
on the same right line ; through these points let there pass 
two difierent systems of parallels, intersecting each other ; 



10 SYSTEMS OF LINES. 

then, a right line passing through any one of their intersec- 
tions, and parallel to the first right line, will pass through 
the next consecutive intersection. 

See Fig. 10, where A, B. C, etc., are at equal distances on the 
line AE. Through these pass the parallels AO, BL, CM, etc., 
and also the parallels AP, BQ, CR, etc. H, K, etc., are the 
intersections of the two S3'stems. Now, a line passing through 
H, and parallel to AE, will pass through the next intersection, 
K. For, let the point in which such a line meets the line CR be 
denoted by X, and that in which it meets BL be denoted by Y. 
Then, since HY and AB are parts of parallels, included between 
the parallels AO, BL, they are, equal. Also HX and BC are 
equal, being parts of parallels included between the parallels BQ 
and CR. But BC was taken equal to AB .-. HX = HYs that is, 
the points X and Y, in the same direction from H, are at the 
same distance from it, or they are the same point, and therefore 
identical with K, since that is the only point situated on both the 
lines CR and BL. 

[It is here assumed that two lines can have only one point in 
common. This is an obvious consequence of the definition of a 
right line. For to suppose that the two lines CR and BL, passing 
through K in two different directions, could meet at some other 
point, would involve the absurdity that this second point could be 
at once in two different directions from K.] 

If the line passing through H, and parallel to AE, passes through 
K, it will for the same reason pass through I, etc., etc. 

37. If any point be taken on one of two intersecting lines, 
its distance from the other line, measured in any given direc- 
tion, will be in a constant ratio to its distance from the inter- 
section of the two lines . 

Thus, in Fig. 11, a, DE is the distance of a point, E, from the 
line AB, measured in a given direction, PQ, and AE is the dis- 
tance of the same point from the intersection A ; and it is to be 

DE . . , 

proved that the ratio ^=r— is constant^ i.e., that it is the same 
AE 

wherever on the line AC the point E may be taken. 



SYSTEMS OF LINES. 11 

On this line AC let a number of poin_ts_ (F, G, H, W, etc.) be 
taken so that the distances AF, FG, GH, etc., shall be equal. 
(See Fig. 11, b.) Through these points let the system of paral- 
lels IF, JG, KH, etc., pass, in the given direction of measure- 
ment, PQ. Also through the same points let lines FM, GN, etc., 
pass parallel to AB. Through the points I, J, K, etc., where the 
line AB intersects the former system, let lines pass parallel to 
AC. They will pass through the intersections S, T, U, Y, etc. 
(§ 36.) 

Now, since IF, JS, are parts of parallels included between the 
parallels AB and FM, they are equal. And since IF, SG are 
parts of parallels included between the parallels AC and IV, they 
-also are equal. In the same way, KU, UT, TH, LY, YX. XV, 
VW, etc., may each be proved equal to IF. Therefore JG = 2 
xIF, KH — 3 X IF , "LW = 4 X if, etc. ; . •. the ratio of the dis- 
tances from the line AB and the point A is the same for each of 

^ . . IF _ JG 2 xlF 

these points F, G, H, etc., viz., -t=i — For =i =n 

AF AG 2 X AF 

IF KH 3 xlF IF ..T , 1 

etc. Now, the equal 



AF AH 3 X AF AF 

distances AF, FG, etc., between the successive points A, F, G, 
etc., might have been taken at any value either greater or less than 
the one actually employed. Hence, they ma}^ be taken at such a 
value that one of the points, F, G, etc., will fall on any point on 
the line AC that may be assigned, or we may regard them as so small 
that these points should coincide with all the successive points of 
the line, yet the demonstration would not be impaired. Hence, 
what has been proved of these points must be true of every point 
of the line AC, viz., that its distance from the given line, AB, 
measured in the given direction, PQ, has a Jixed ratio to its dis- 
tance from the intersection A. 

'68. For any given direction of measurement, it is evident that 
the magnitude of this ratio will depend simply upon the angle 
between the two given lines. For at any given distance from the 
intersection of two lines, their distance apart will depend upon the 
amount of their divergence. This is expressed in the words, " the 
ratio is a function of the angle.*' 



12 FUNCTIONS OF ANGLES. 



FUNCTIONS OF ANGLES. 

39. When the value of one quantity depends upon the value of 
another, the former quantity is called a function of the latter. 
Thus, the quantity (10 a'^ + ^) is ^ function of a. There are 
certain functions of angles which are used with great advantage in 
calculations where the use of the number directly expressing the 
magnitude of the angle (in degrees, etc.) would be extremely 
inconvenient, or even impossible. Those functions in most 
common use are the sine, tangent, cosine, and cotangent of 
angles. 

40. When two lines which form an angle are intersected 
by a third line which is perpendicular to one of them, the 
ratio of that part of the third line which is included between 
the other two to its intercept on the line to which it is per- 
pendicular, is called the tangent of the angle. 

41. The ratio of the same part of the third line to its 
intercept on the line to which it is not perpendicular, is 
called the sine of the angle. 

Thus, in Fig. 12, when the lines AM, AN, including the angle 
a, are intersected by the line BC perpendicular to AM, the ratio 

—^m- is the tangent of the angle a. This ratio is the same at 
AK 

whatever point the line BC may meet AM, provided only it be 

perpendicular to AM (§ 37), for that fact fixes the direction in 

which the distance KH is measured as one differing 90° from that 

of AM. Also (§ 38), this ratio is a function of the angle a. 

TTTT 

The ratio - is the sine of the angle a. It is the same at 

AH 

whatever point BC may meet AN (§ 37), and is a function of the 

angle a. (§ 38.) 

42. The cosine of an angle is the sine of the complement 
of that angle. The cotangent of an angle is the tangent of 
the complement of that angle. 

The prefix "co" is here an abbreviation of "complement." 



FUNCTIONS OF ANGLES. 13 

43. For use in formulae, etc., the names of these four functions 
are abbreviated into tan^ sin^ cos, and cot. Thus, tan a is read 
"the tangent of a," sin 25° is read ''sine of twenty-five degrees." 
When one of these abbreviations is followed by the sign (-) the 
angle is indicated in terms of its function. Thus, cos" J is "the 
angle wliose cosine is one-half;" cot "3, "the angle whose cotan- 
gent is three." 

44. In Fig. 13, if AB and AD are perpendicular to each other, 
the angle /? is the complement of a. Through any point, P, on 
the line AC let a line pass parallel to AD and another parallel to 
AB. The former will be perpendicular to AB and the latter to 

.^ ..^.. r^. ^^ MP . NP 

AD. (§ 26.) Then -^r^ r= tan a, ^ sin a, — zzir- = tan 

AM AP AN 

NP — 

13 =z cot a, and — =r- z= sin /5 z= cos a. Now, AM =: NP. 
AP 

NP ATvr 

(§ 35.) ••. ;= ' the ratio of the intercepts of PM on 

AP AP 

AB and AC. Hence we may give a new definition of the cosine, 

as follows ; 

When two lines, which form an angle, are met by a third 
line, which is perpendicular to one of them, the ratio of its 
intercept on that line to its intercept on the other is called 
the cosine of the angle. 

45. Since AM = NP and AN _ mP (§ 35), cot a = tan /? = 

NP A^f — 

= ■» the ratio of the intercept AM to the part MP of 

AN MP 

the intersecting perpendicular. Hence we may give the following 
definition of the cotangent: 

When two lines, which form an angle, are met by a third 
line, which is perpendicular to one of them, the ratio of its 
intercept on that line to that part of the intersecting per- 
pendicular included between the first two lines is called the 
cotangent of the angle. 



14 FUNCTIONS OF ANGLES. 

These two definitions may be applied, as well as the analogous 

tFtt tt-tt 

ones of §§ 40, 41, to Fig. 12, where we have -z==— = sin a, 



AH AK 

AK AK 

= tan a, z= cos a, = cot a. 

AH KH 

46. All these definitions are applicable, not only to angles less 
than 90°, but to angles from 90° to 180° as well. In the latter 
case, however, one of the lines forming the angle must be con- 
sidered as extending beyond the point of intersection. Thus, in 

KH 

Fig. 14, where /5 is the angle between AM and AN, -^i^- =^ sin. 

AH 

KH . AK , AK 

/?, = tan /5, = cos /?, and - = cot (3. 

AK AH KH 

Moreover, though, as we have already seen (§ 20), an angle of 
180° is the greatest possible divergence between two lines, yet it 
is very common to speak of angles of from 180° to 360°, or even 
of angles of any number of degrees whatever. This usage is the 
natural and even necessary result of the simple process of the 
addition of angles. Thus, in Fig. 15, if the angle a= 110° and 
jS = 100°, it is natural to speak of their sum as an angle y, equal 

KH 

to 210°. And here, exactly as in the former cases, - =z sin 

AH 

KH AK , AK ^ 

y, = tan r, ■ = cos r, and ■ =z cot r. 

AK AH KH 

The definitions of § 41 are no less applicable to these cases- 
than those of §§ 44, 45 ; as the student will see when he reaches- 
the discussion of negative angles. 

47. When two lines perpendicular to each other are inter- 
sected by another line, the square of that part of the latter 
included between them is equal to the sum of the squares of 
its intercepts on the first two lines. 

In Fig. 16, let there pass through the intersection of FE and 
CD (which are perpendicular to each other) a line perpendicular 
to AB, and cutting it in K, so as to divide the distance GH or s 



FUNCTIONS OF ANGLES. 15 



into two parts, GK or p, and KH or q. Let the distances GO and 
HO be called m and n. Then it is to be proved that 
s' = m'+ii2. (See § 10.) 

Cos a =z J^5 and cos a = — » .•.-!- = — , and ps = ni2. (i.\ 

In the same way, — 

cos Sz=z% and cos jS =: -, .*. 51, z=z -^ and qs =z 112. (2.) 

Adding (1) and (2) we have 

ps + qs = m^ + 112, 

or s (p + q) = m' + n'. 

But p-|-qz=s, .-. s'=:m^+n^ 

48. Dividhig this equation through by s^, we have 

_ 1112 112 /ii\' I /na\^ 1 

1 = T' or (-)+( — )=1- 

But — = sin a and — = cos a, .'. sin^ a -4- cos'* a = 1, 

IS s 

(These expressions are read, "the square of the sine of a," etc.) 

Hence the sum of the squares of the sine and cosine of any 
angle is equal to unity. 



49. The ratio -^=:=r^ (Fig. 12), is equal to ■ — ■; or 

AK AH AH 

in general terms : ■ — 

The tangent of an angle is equal to the sine divided by 

, , . , sin a 

the cosine ; or tan a = . 

cos a 

1 sin a 

Hence sin a = cos a tan a, and cos a = ; • 

tana 



.r, rp, ,. AK AK . KH 

oO. The ratio • = - . -7- ; or 

KH AH AH 

The cotangent is equal to the cosine divided by the sine ;; 

cos o 

or cot a = . 

sin a 



16 FUNCTIONS OF ANGLES. 



Hence cos a = sin a cot a, and sin a = — 1— . 

cot a 

These results follow from § 49, for, if tan a = — -, 

cos a 

*-\.r.^ 4^^^ /OAO \ sin (90° — a) cos a 
then tan (90 —a) = -— - — ^ ; or cot a = . 

cos (90 — a) sin a 

51. The ratio -^- = 1 — ^ ; or, 

KH AK 

The cotangent is the reciprocal of the tangent ; 

i. e., cot a =-. i 

tan a 

whence tan a cot a = 1' and tan a = . 

cot a 

This may be obtained by comparing §§ 49, 50; for if 

J- S"^ " J ^ cos a . . ^, ^ 1 

tan a = , and cot a = —, , evidently cot a = 

cos a sin a '' tan a 

52. By means of his square, with graduated edges, the pupil can now solve 
the following problems : 

A. To measure the sine or cosine of a given angle. 

Selecting a convenient number of units, such as 10, lay off the distance on 
one of the lines forming the given angle, and from the point thus fixed draw a 
perpendicular to the other line. Then measure the distances named in the 
definitions (§§ 41, 44), and divide by the assumed number of units. 

B. To measure the tangent of a given angle. 

Select a convenient number of units, as before, and lay it oflf from the inter- 
section of one of the lines, then draw a perpendicular to that line at the point 
thus fixed, and measure the part included between the given lines. Divide 
this length by the selected number of units. 

C. To measure the cotangent of an angle, the most convenient way will be 
to measure its tangent and take the reciprocal. 

D. To construct, at a given point in a given line, an angle having a given 
sine or cosine. 

Lay off on the given line from the given point. A, a distance, AB, equal to 
the denominator of the given ratio. Take on one edge of the square a distance 
equal to the numerator of the same, and if the ratio be a sine, fix this point of 
the square at the point B, but if the given function be a cosine, at the point A, 
and turn the square until the remaining edge passes through the other point. 
Then draw a line through A, making the required angle. See Fig. 17, where 
a zz: sin- ^. 

If the distances indicated by the terms of the fraction are inconveniently 
large or small, the fraction should first be reduced, by dividing or multiplying 
both nunier;itor and denomi;ui;or alike. 



FUNCTIONS OF ANGLES. 17 

E. To construct an angle whose tangent or cotangent is given. 

Lay off on the line, as before, a distance equal' to the denominator of the 
ratio (reduced, if convenient), if the ratio be a tangent, but if the cotangent is 
given, lay off the numerator. At the point thus fixed erect a perpendicular 
equal to the remaining term of the ratio, and connect its extremity with the 
given point. 

Examples. 

1. Construct the angle whose sine is J-, that whose tangent is 2, 
one whose cosine is f , and one whose cotangent is |-. 

2. Construct cos" i, tan" f, sin' f, cot* 3, sin* f, cos- f, 
tan-;^, cos- -^-^, cot" -f'-^. 

The teacher may multiply these examples indefinitely, and 
should provide examples for the converse rules, of measuring the 
functions of given angles. 

3. Compute the cosine of the angle whose sine is f , the tangent 
of the angle whose cotangent is f . 

4. Compute all the other functions of cos- f , of tan- -f>-^. 

In the last example we easily find cot = ^-. To find the cosine 
we have from § 49, sin a = cos a tan a, .*. sin"^ a == cos- a tan'-* a, 
that is, in this case (since tan a =xV' ^^^^ hence tan^ a =t4^4), 
sin2 a = -^^ cos^ a. Now, let x represent cos2 a, then -f^^ x = 
sin"'' a, and (§ 48) x + -^-^^ cc = 1, or ^| x= 1, whence x =i||, 
the square of the cosine of a. Hence, the cosine of a = |f, and, 
since sin a = cos a tan a, sin « = yf • tV = tV* 

5. Compute the other functions of cot- |^, of cos" y^g. 

After performing examples 3, 4, 5, construct all the angles from 
the given functions, and measure the required functions to prove 
the work. 

6. On each of two lines, making right angles with each other, 
is measured from their intersection a distance of 100 feet. What 
is the distance between the two points thus fixed? (See § 47.) 
Answer: 141.42+. 

7. If a staff 4 feet long, held perpendicularly, casts a shadow 
3 feet long, what is the distance from the top of the staff to the 
end of the shadow? 

8. My brother's house is due north from mine, and due west 
from my cousin's. My house is f mile distant from my cousin's, 
and I mile from my brother's. What is the distance from my 
brother's house to my cousin's? 



18 FUNCTIONS OF ANGLES. 

9. A may-pole 32 feet high, standing on a level plain, is broken 
partly through, so that the top rests on the gronnd 16 feet from 
the foot of the pole. How far above the ground is it broken? 

[Let X = the length of the part broken off, and y = the length 
of the remainder (the distance required). Then we may obtain 
(a; -J- y) (x — 2/) == 256, whence, as a; -|- 2/ i^ known, y is readily 
found. Answer: 12.] 

10. From the top of the mast of a sail-boat a rope is stretched 
to a point on the deck 9 feet distant from the foot of the mast. 
The rope is three feet longer than the mast. How tall is the 
mast? 

11. The village A is due north from B, and B is due west from 
C, to which straight roads extend from both A and B. Two trav- 
elers, the one from A, the other from B, met at C ; and, on com- 
paring the distances travelled, it was found that one had come five 
miles further than the other, and that the entire distance traveled 
by both was three times the distance, in a direct line, between 
the starting points. How far had each traveled? 

12. A township, 1,920 rods square, is divided into sections by 
lines parallel to the boundaries, at distances of 320 rods. Re- 
quired, the distances between points in the township to be located 
by the teacher. 

53. If a line meet a system of convergeiits, the sines of 
the angles which it makes with any two of them are recip- 
rocally proportional to its intercepts upon them. 

Thus, let PM, PN, and PO (Fig. 18), be a system of conver- 
gents intersected by the line AX, which makes, with PM, the 
angle a, with PN the angle jS, and with PO the angle ;'. Then 
will these proportions be true, viz.. 

Sin a : sin(3 : : PN : PM, 
sin a : sin ;- : : PO : PM, 
aiid sin /3 : sin ^ : : PO : PN. 

For, through P, the point of convergence, let the line PR pass 
perpendicular to AX, and meeting it at R. Then, 

PR . ^ PR , . PR 

sni a = , sni p = ■? and sm / = - 

PM PN PO 



DETERMINATION OF POINTS IN A PLANE. 19 

Hence, PR = PM sin a = FN sin i3 = PO sin ;'. 

Now, from the equation 

PM sin a = PN sin 13, 
may be derived sin a : sin jS : : PN : PM ; 
from PM sin a = PO sin ;', > 

comes sin a : sin ^^ : : PO : PM, , 

and from the equation 

PN sin i3 = PO sin y, 
we have sin ^3 : sin ;' : : PO : PN. 

The pupil may here urge the objection that the theorem is not 
sufficiently definite. For ''the angle which AX makes with PM" 
might be the angle a ' as well as a. Which of two such angles is 
intended? 

Answer: Either; for the sines of a and a are identical, viz., 

PR 

., although the angles are different. (§ 46.) Hence the 

PM 

statement should not be more explicit. 

That sin a and sin a' are identical in algebraic sign as well as in numerical 
value will appear hereafter. (§ 89.) 

DETERMINATION OF POINTS IN A PLANE. 

54. The position of a point in a plane is determined when 
its distance and direction from a known point are given. 

This is the most obvious and natural way of describing the position of a point,, 
and examples of it in daily use are abundant. 

55. Direction from the known point is given by compari- 
son with a known direction. 

Thus, (Fig. 19,) the position of P is determined if we know 
that it is at a distance of one inch from the known point, A, on a 
line, AQ, which makes an angle of 30° above the line AX, whose 
direction is known. 

56. The quantities which must be given, in order to 
determine the position of a point, are called the coordinates 
of that point. 



20 DETERMINATION OF POINTS IN A PLANE. 

The coordinates of P, (Fig. 19,) are the distance r = 1 inch, and the angle 
^=30°. 

57. This method of determining the position of a point, 
where the coordinates are a distance and an angle, is called 

The Method of Polar Coordinates. 

58. The known point, A, is called the^o/e, and the line 
AX, passing through the pole in the known direction, is the 
initial line, 

59. The distance r is called the radius vector, and the 
angle d^ the vectorial angle. These two are i\\Q polar coordi- 
nates. 

60. By general consent, the initial line is always described 
from the pole, toivard the right. The radius vector is also 
always considered as described from the pole, and the vec- 
torial angle is measured continuously around the pole from 
right to left, — i. e., in a way contrary to the motion of the 
hands of a watch. 

Thus, (Fig. 20,) if the hne AB makes an angle of 45° with AX, 
and AC makes an angle of 45° with AB, and so on ; the vectorial 
angle for any point on the line AB is 45° ; on the line AC, 90° ; 
on AD, 135° ; on AE, 180° ; on AF, 225° ; on AG, 270° ; and on 
AH, 315°. The vectorial angle for points on the line AX may be 
considered either 0° or 360°. We may continue, and regard the 
line AB as corresponding to a vectorial angle of 3G0° -|- 45°, or 
405°, as well as of 45°, while the value of for a point on AC is 
either 90° or 450°, etc., and so the circuit may be repeated as 
many times as we please. That is. 

The direction of the radius vector is not altered when the 
vectorial angle is either increased or diminished by 360°, or 
any multiple of that quantity by a whole number. 

Gl. Problem. — To locate the point whose coordinates are 
r ^ m, ^ = (« -f- /3) ; also the point whose coordinates are 
r = m, ^== {a — (3). 

For the solution of either part of this problem it is obvious that 



DETERMINATION OF POINTS IN A PLANE. 21 

two lines must be found passing through the pole, the one making 
an angle equal to a above the initial line, the other making an 
angle of ^ with the former. But in the first case, since the sum 
of two angles is required, the angle jS must be taken on the upper 
or left-hand side of the first line drawn, while in the second case 
it must be taken on the lower or right-hand side of that line, so 
that the inclination of the second line shall be less than that of the 
first by j3. In either case, a distance equal to in must be measured 
from the pole on the second line, and its extremity will locate the 
required point. 

From this example it will be seen that the negative sign indi- 
cates a reversed direction of measurement^ for when (9 = « -|- /^ we 
measure both a and (3 from right to left, but when 6 = a — /3, the 
latter is measured from left to right. Obviously, if j3 exceeds a 
in numerical value, the hne on which the radius vector is to be 
measured will fall below the initial line, but in this case the value 
of the expression a — j3 becomes negative, hence the formula 
6 = — Y indicates that a vectorial angle equal to y is to be 
measured helow the initial line. In all cases 

Negative angles are measured from left to right, as posi- 
tive angles are measured from right to left. 

62. Any negative angle may be rendered positive by adding to 
it 360° a suflScient number of times, and any positive angle greater 
than 360° may be made less than 360° by subtracting that quantity 
or a sufficiently large multiple of it, and since these operations 
do not affect the direction of the radius vector (§ 60), it follows 
that 

Any point whose vectorial angle is negative, or positive 
and greater than 360°, may be denoted by a positive vectorial 
angle less than 360°, without change of the radius vector. 

63. Problem. — To locate the point whose coordinates are 
6 = a^ r = sn -|- 11 ; also the point whose coordinates are (? =«, 
r = iii — n. 

Having found, as before, a line making an angle equal to a above 
the initial line, we measure upon it from the pole a distance equal 
to m. Both the points in question are upon this Hne, and at a 



22 DETERMINATION OF POINTS IN A PLANE. 

distance of n from the point already located, but it is obvious 
that when the radius vector is the sum of m and n, the measure- 
ment of n should be continued in the same direction in which m 
was measured; but when the difference is sought, the measure- 
ment should be in the contrary direction. If n is numerically 
greater than m, the point will fall beyond the pole. Hence as 
before, the negative sign indicates a reversed direction of measure- 
ment; and 

Negative radii vectores are measured in a direction con- 
trary to that indicated by the vectorial angle. 

64. The principle illustrated in §§ 61, 63, is clearly a 
universal one, and it will hereafter be assumed that when- 
ever measurement in a determinate direction is considered 
as affording a positive result, that which is measured in the 
opposite direction is of necessity regarded as negative. 

65. The difference between two opposite directions is 180°; 
hence a change of sign in the radius vector produces the same 
effect as an increase or decrease of the vectorial angle by 180° ; 
and if both these operations are performed at once they must neu- 
tralize each other. Therefore, 

Any point whose radius vector is negative may be denoted 
by a positive radius vector of the same length, if the vec- 
torial angle be increased or diminished by 180°. 

Examples. 

Locate the points whose coordinates are as follows, 
r =4, ^ = 90°; r = 3, 6 = sm-^',v = 5, ^ = tan-2;r = |, 6 = 
cot- f ; r = |-, = cos- f, etc., etc. 

Reduce the following coordinates to positive ones, in which 
6 < 360°, and locate the points, r = 5, ^ = — 90° ; r = — 3, 
6 = 270° ; r = 3, ^ = — 270° ; r = — 3, ^ = — 450°, etc. 

66. While for some geometrical uses this " method of polar 
coordinates" is very serviceable, and, indeed, essential, in gen- 
eral its application is inconvenient, because the use of both dis- 
tances and angles is necessary throughout. Hence another 



DETERMINATION OF POINTS IN A PLANE. 23 

method of determining the position of a point is more frequently 
employed, called, from the name of its inventor, Descartes, 

The Cartesian Method of Coordinates, 

In this method all the quantities employed are distances, and 
the position of a point is determined by reference to iwo inter- 
secting right lines^ the position of which is known. These lines 
(AX and AY, Fig. 21,) are called the axes, and the point (A) at 
which they intersect is called the origin. 

67. The coordinates of a point in the Cartesian method, 
are its distances from the two axes, the distance from either 
axis being measured on a line parallel to the other. Thus, 
RP, QP, are the coordinates of the point P. Since 
KP = AQ and QP = AR (§ 35) either coordinate may be 
measured on one of the axes, from the origin to the inter- 
section of a parallel to the other axis. 

Let the student notice that the coordinates are always measured 
from the axes, or from the origin, never toicard them. 

68. One of the axes is always considered as passing 
through the origin toward the rigid, and the coordinate 
measured on this axis, or parallel to it, is called the abscissa, 
the axis itself being therefore called the axis of abscissas. 
Thus, AX is the axis of abscissas, and AQ or RP is the 
abscissa of P. 

69. The coordinate measured on the other axis, or parallel 
to it (AR or QP) is called the ordinate, and this axis (AY) 
is named the axis of ordinates. 

70. Abscissas are usually denoted by the letter x, and 
ordinates by the letter y ; hence the axis of abscissas is 
frequently called the axis of X, and the axis of ordinates, 
the axis of Y. 

71. Abscissas measured toward the right are considered 
positive, hence, (§ 64) those measured to the left must be 
considered negative. Ordinates measured upward are posi- 
tive, hence those measured downward are negative. 



24 DETERMINATION OF POINTS IN A PLANE. 

Thus, in Fig. 21, the coordinates of P are both positive ; those 
of P" are both negative ; for P' the ordinate is positive, and the 
abscissa negative ; and for P'" the abscissa is positive, but the 
ordinate is negative. 

It is obvious that all points cannot be denoted in the Cartesian 
as in the polar metliod, b}^ the use of positive coordinates only. 

72. The two axes divide the plane in which they are situated 
into four parts, of which that situated above the axis of X and to 
the right of the axis of Y is called the first angle; that above the 
axis of X and to the left of the axis of Y, the second angle; that 
below the axis of X and to the left of the axis of Y, the third 
angle; and that below the axis of X and to the right of the axis 
of Y, the fourth angle. 

Hence, for points in the first angle, x is -|- and y is -|- ; in the 
second angle, x is — and y is -j- ; in the third angle, x is — and 
y is — ; in the fourth angle, x is -[- and y is — . 

73. The angle between AX and AY, denoted by the 
letter a», is of course a known angle, since the position of 
both axes is known. It may have any value from 0° to 
180°. When it is a right angle, the Cartesian method 
becomes the method of rectangular coordinates, otherwise it 
is the method of oblique coordinates. 

For most geometrical uses, the processes of calculation are 
much simpler when rectangular coordinates are used than when 
they are oblique. Hence, in the following §§ the axes emplo3'ed 
will always be rectangular, unless the contrary is stated. 

Examples. 

Locate the points whose coordinates are as follows : 
X=3, y = 2; x = 3, y = — 4; x= — 2, y = 5; x = 0, 
y=3;x=— 2, y=0;x= — 2, y = — 3, x=0, y = 
^ 2 ; X = 0, y = 0, etc., etc. 

74. Since the position of a point is completely determined by 
either the method of polar or of Cartesian coordinates, it must be 
possible to find an expression for the value of the coordinates 
of any point in one of these systems in terms of the coordinates of 

/ 



DETERMINATION OF POINTS IN A PLANE. 25 

the same point in the other system ; i.e., to translate the expres- 
sions of one system to those belonging to the other. 

75. Problem: To find expressions in polar coordinates 
for X and y, when the pole coincides with the origin of 
rectangular Cartesian coordinates, and the initial line with 
the axis of abscissas. 

In Fig. 22 we have, from § 44, 

cos ^ == -^^^, = - • .*. x=rcos^ 

AP *• 

OP T 

and, from § 41, sin d = - _ = ^ » .• y = !• sin ^; 



AP 



r 



the formulae required. 



76. Problem : To find expressions in rectangular coordi- 
nates for r and ^, the position of the pole and initial line 
being as above. 

In the same figure (§§ 40, 45) 

tan^ = -S£- = ^ and cotl9 = -^=?. 

AQ ^ QP y 

V X 

Hence, 6= tan* ^- = cot- -• 

X y 

Also (§ 47), AP^ = AQ^ + QP^ 

therefore, r = V' x'^ + J'^- 

11. We have seen that all points may be denoted by positive 
coordinates in the polar system, while for some points the Carte- 
sian coordinates must be negative. Hence, if the formulae of § 75 
are to determine anything more than the numerical values of x and 
y, we must consider sin d and cos 6 (and therefore, of course, 
tan d and cot 0) as affected with positive or negative signs, which 
will depend upon the magnitude of d. And these functions are, 
therefore, universally assigned their algebraic signs, in accordance 
with the following four formula (deduced above), viz: 

sin = ^, cos = - , tan =T , and cot 6 = ?. 
i' 1- X y 



26 SIGNS OF THE FUNCTIONS OF ANGLES. 

In these formulse, r is always considered positive (§ 65), which, 
it will be seen, is entirely consistent with the formula of § 76, 
r = \/ x:^ + y% since, whatever the sign of x or y, their squares 
are positive, and the square root of the sum of these squares may 
be taken as positive. 

SIGNS OF THE FUNCTIONS OF ANGLES. 

78. The sine of a positive angle less than 180° is positive, 
of an angle between 180° and 360° negative. 

Since the Cartesian axes here used are rectangular, the value of 
d for any point in the first angle is between 0° and 90°; in the 
second angle, between 90° and 180° ; in the third, between 180° 
and 270° ; and in the fourth, between 270° and 360°. Now, since 

sin d = — and r is positive, sin 6 must have the same sign as y, 

which is positive in the first and second angles, and negative in 
the third and fourth. (§72.) 

79. The cosine of a positive angle less than 90° is posi- 
tive, of an angle betw^een 90° and 270°, negative, and of an 
angle between 270° and 360°, positive. 

X 

Since cos = — , cos 6 must have the same sio-n as x, which is 
J,' 

positive in the firsfc and fourth angles, and negative in the second 
and third. 

80. The tangent and cotangent of a positive angle less 
than 90° are positive, of an angle between 90° and 180°, 
negative, of an angle between 180° and 270, positive, and 
of an angle between 270° and 360°, negative. 

v X 

Since tan 6 = — and cot = — , these functions will be posi- 
X y 

tive when x and y have like signs, and negative when their signs 

are different. But from § 72 we see that the signs of x and y 

are alike in the first and third angles, and unlike in the second 

and fourth. 



SIGNS OF THE FUNCTIONS OF ANGLES. 27 

81. Any function of an angle will be in all respects equal 
to the same function of that angle increased or diminished 
by 360°, or by any multiple of that quantity by a whole 
number. 



sin 


sin 




tan 
cos 


. = tan 
cos 


- ((9 db n 360°) 


cot 


cot J 





That is, 



where n is an integer. 

For supposing v to retain the same value, the point indicated is 
the same (§ 60), hence, the values of r, x, and y, will all be un- 
changed, numerically and algebraically. 

82. Any function of an angle is numerically equal to the 
same function of that angle increased or diminished by 
180°. 

See Fig. 22, where it is evident that, since the divergence of the 
line AB' from AX' is the same as that of AB from AX, the values 
of r, X, and y will be numerically equal for two points situated 
at equal distances from A, one on each line. (§ 38.) Hence the 
ratios of these quantities will be numerically equal. 

83. If two angles differ by 180°, their sines will have 
different signs, also their cosines ; but their tangents and 
cotangents will have like signs. 

For if the smaller angle is between 0° and 90° the larger will be 
between 180° and 270° ; if the smaller is between 90° and 180° the 
larger will be between 270° and 360° ; if the smaller is between 
180° and 270° the larger will be between 360° and 450°, and its 
functions will be the same as those of an angle between 0° and 
90° ; if the smaller is between 270° and 360° the functions of the 
larger are those of an angle between 90° and 180°, etc. Now, 
from §§ 78-80, it will be seen that the sine of the larger angle is 
negative whenever that of the smaller is positive, and vice versa, 
that the same relation exists between their cosines, and that when 
the tangent of one is positive, the other is so also, etc., etc. 



28 SIGNS OF THE FUNCTIONS OF ANGLES. 

84. Combining the results of §§ 82, 83, we may write the 
formulae, 

sin (180°+ a) = — sin a, cos (180°+ «) = — cos a, 

tan (180°+ a) = tan a, cot (180°+ a) = cot a. 

85. Any function of a negative angle is numerically equal 
to the same function of an equal positive angle. 

In Fig. 22 it is evident that the divergence of AB from AX is 
the same as that of AC from AX, and if two points are taken, one 
on each line, at equal distances from A, the value of r, x, and y 
are numerically equal for the two points, (§ 38,) whence the 
proposition follows. 

^Q. The cosine of a negative angle has the same sign as 
the cosine of an equal positive angle ; but the sine, tangent, 
and cotangent of a negative angle have the opposite signs 
from the corresponding functions of an equal positive 
angle. 

If ihe positive angle be between 0° and 90°, the negative angle 
will have tlie snme functions as an angle between 270° and 360° 
(§ 81) ; if the positive be between 90° and 180°, the negative will 
have the functions of an angle between 180° and 270°, etc., etc. ; 
and on applying the principles of §§ 78, 80, it will be seen in 
every case that the cosines of the positive and negative angles 
agree in sign, but that when the sine of either is negative, that of 
the other is positive, and vice versa; and that their tangents and 
cotangents also have opposite signs. 

87. By comparison of §§ 85, 86, we may write: 

sin ( — a) = — sin a, tan ( — «) = — tan a, 

cos ( — a) = cos a, COt ( — a) = — COt a. 

88. Any function of — a equals the same function of ( — a 
+ 360°), or (360° — a), (§ 81,) hence, 

sin (360°— «) = — sin a ; tan (360° — «) = — tan a ; 

cos (360° — «) = cos a ; and cot (360° — a) = — cot a. 
89. If i3= 180°+ «, sin (180° — a) = sin [360°— (180°+ a)] 



SIGNS OF THE FUNCTIONS OF ANGLES. 29 

= sin (360° — /^) = — sin/? = — sin(180° + a) =— (— 
sin a) z= sin a, (See §§ 88, 83.) 
Also, cos (180° — a) = cos [360° — (180° + a)] = cos (360 

_/3) = cos /5 = cos (180°+ a) = — cos a. 
In the same wa3% 

tan ( 180° — «) = tan (360° — /3) = — tan /3 = — tan (180° 
-)- a) = — tan a. 

And cot ( 180° — a) = cot (360° — /3) = — cot /3 == — cot ( 180° 

-|- a) = — cot a. 
This result may be thus stated in words : 

Any function of an angle is numerically equal to the same 
function of its supplement, and the sines of two angles 
which are supplements have like signs, but their tangents, 
cosines, and cotangents have unlike signs. 

90. From the definition of § 42 we have, 

sin (90° — «) = cos a, 

and tan (90° — «) = cot a; 

wlience, cos (90° — «) = sin a, 

and cot (90° — «) = tan a. 

yi. 90° + a is the supplement of 90° — a; hence (§§ 89, 90), 
sin (90° + a) = sin (90° — a) = cos a; 
tan (90° + «)== — tan (90° — a) = — cot a ; 
cos (90° + a) = — cos (90° — a) = — sin a ; and, 
cot (90° 4- a) = — cot (90° — a) = — tan a, 

92. Sin (270° — «) = sin [90° + (180° — a)] = (§ 91), cos 

(180° — ./.)=(§ 80) — OOSr,; 

tan (270° — a) = tan [90° + (180° — a)] = — cot (180° 

— «) = — ( — cot a) = cot a ; 

cos (270°— a) = cos [90°+ i^ioj' — a)] = — sin (180° 

— «) = — sin «; 

cot (270° — a) = cot [90° + ( 180° —«)]= — tan ( 180° 

— «) = — ( — tan a) = tan a. 

93. Sin (270° + a) = sin [90° + (180° + a)] = (§ 91), cos 

(180° + «) = (§ 84), — cos a; 
tan (270° + a) = tan [90° + (180° +«)]= — cot (180° 
-j- a) = — cot a ; 



30 



SIGNS OF THE FUNCTIONS OF ANGLES. 



COS (270° + a) = cos [90° + (180° + a)] = —sin (180'' 

+ a) = — ( — sin a) = sin a ; 
cot (270° + a) = cot [90° + (180°+ a)] =— tan (180^ 

+ a) = — tan a. 

94. The results of §§ 78 to 93 are collected in the following 
tables, which the student should carefully memorize, and ma^^ 
extend indefinitely by the principle of § 81: 

Table I. 





0° to 90° 


90° to 180° 


180° to 270° 


270° to 360° 


sine 
tangent 
cosine 
cotangent 


+ 
+ 
+ 
+ 


+ 


+ 
+ 


+ 


Table II. 


a 


90°— « 


90°+a 


180°— « 


180°+^:^ 


270°— a 


270°+a 


360°— « 


360°+a 


sin a 
tan a 
cos a 
cot a 


COS a 
cot a 

sin a 
tan a 


COS a 
— cot a 
— sin a 
— tana 


1 

sin a 

— tana 

— cos a 

— cot a 


— sin a 
tan a 

—cos a 
cot a 


— COS a 
cot a 

— sin a 
tana 


— COS a 
— cot a 

sin a 
— tan a 


—sin a 

— tana 

cos a 

— cot a 


sin a 
tana 
cos a 
cot a 



In memorizing, the student will be aided by noticing that the 
signs in the first two columns of Table II. correspond with those 
in the j^rs^ column of Table I., in the next two, to those in the 
second column, etc. ; also, that the function changes name (from 
sine to cosine, etc.) when the angle to which rh « is added is a 
multiple of 90° by an odd number, but does not change when that 
angle is a multiple of 90° by an even number. 

From § 71 we see that the extension of either table in either 
direction will be simply a repetition, in the same order, of four 
columns in the case of Table I. and of eight in the case of Table 
11. When reference is to be made to these tables in future, they 
are to be considered as extended in this manner if necessary. 



SIGNS OF THE FUNCTIONS OF ANGLES. 31 

95. From Table I. it appears 

(a.) That like functions of two angles must have the same sign 
unless some multiple of 90°, (including 0°, which = X 90°,) is 
intermediate in value between the angles. Also, 

(6.) From Table II., that a sine numerically equal to sin a. 
belongs to one angle between 0° and 90°, to one between 90° and 
180°, etc., also to one between 0° and — 90°, etc. ; and that a sine 
numericall}^ equal to cos a likewise belongs to an angle between 0°' 
and 90°, to one between 90° and 180°, etc. ; and that tangents, 
cosines, and cotangents are similarly distributed ; whence 

(c.) It follows that any function of any positive or negative 
angle whatever is equal to the like function of some angle between 
the limits of 0° and 90°, inclusive ; which angle may be found by 
adding or subtracting the requisite multiple of 90°. 

96. Table II. is complete^ i. e., it embraces all the angles whose 

sine is numerically equal to sin a or to cos a, whose tangent is 
numerically equal to tan a or to cot a, etc. For, if it be incom- 
plete, there must be at least two angles [see § 95 (6)] such that 
no multiple of 90° is intermediate between them (that is, two 
between 0° and 90°, or two between 90° and 180°, etc.,) having 
like functions which are numerically equal. If so, those functions 
would also be algebraically identical, [§ 95 (a,)] and, by adding 
or subtracting the requisite multiple of 90°, two angles between 
0° and 90° might be found, whose like functions would also be 
be equal. [§ 95 (c.)] Hence, if it can be shown to be impos- 
sible that two angles between 0° and + 90° should have like 
functions which are equal, Table II. is complete. Now, let 6 and 
d' be two positive angles, each less than 90°, made with AX (Fig. 
23) by the two lines AB and AC, and let us suppose that the sines 
(e. g.) of these angles are equal. Let P be any point on AB, and 
through P let a line pass parallel to the axis of abscissas AX, and 
cutting the axis of ordinates at the point R, and the line AC at Q. 
Let X and y, r and be the coordinates of P, and x' and y', r^ 
and d' those of Q. Then, by construction, y = y'; 
and, by hypothesis, 

sin d = sin d' or — = -^ .*. r = r'. 

Now, since P and Q are both in the first angle, x and x' have like 
signs, but 



32 FUNCTIONS OF ANGLES. 



X = V •'^ — y^ ^^^^ x' == V ^''^ — j'^ •*• ^ = x'- 

Hence P and Q coincide ; and thus every point in AB coincides 
with a point in AC, hence 6 = 6'. In like manner it may be 
proved that 6 = 6' it cos 6 = cos 6 \ etc. Hence the proposition 
is proved. 

FUNCTIONS OF THE SUM AND DIFFERENCE OF ANGLES 
AND OF MULTIPLE ANGLES. 

97. The sine of the sum of tv^o angles is equal to the 
product of the sine of the first by the cosine of the second, 
plus the product of the cosine of the first by the sine of the 
second ; i. e., 

sin (a -(- /3) =r sin a cos j8 -f- cos a sin /9. , 

The sine of the difference of two angles is equal to the 
product of the sine of the first by the cosine of the second, 
minus the product of the cosine of the first by the sine of 
the second ; i. e., 

sin (a — 13) z=: sin a cos (i — cos a sin i3. 

The cosine of the sum of two angles is equal to the 
product of their cosines, minus the product of their sines ; 
i. e., 

cos (a -|- /3) = COS a cos (i — sin a sin (3. 

The cosine of the difference of two angles is equal to the 
product of their cosines, plus the product of their sines ; 
i.e., 

cos (« — /3) = cos a cos /3 -|- sin a sin j3. 

These propositions will first be proved on the supposition that 
each of the angles is numerically less than 90° ; and the demon- 
stration will then be extended to embrace all angles. 

The values of the functions of ( — a. — /3) and ( — a-\- (i) may 
be determined by § 87 when we have found the functions of 
(v. -f-i^) and (a — jQ) ; hence we need examine only the cases in 
which the larger angle a is positive. 

In Fig. 24, 25, or 26, let the line AP make the positive angle a 



FUNCTIONS OF ANGLES. 33 

with the line AX, and let the line AQ make the angle (3 with the 
line AP — positive in Fig. 24 or 25, but negative in Fig. 26 — so 
that in the former case the angle made by AQ with AX is equal 
to a -\- (3, but in the latter case to a — (3. Through P, any point 
on AP, let a line, PR, pass perpendicular to AX, and meeting it 
at R, and another, PN, parallel to AX ; also a line, PQ, perpen- 
dicular to AP, and meeting AQ at Q. Through Q let a line, QS, 
pass, parallel to PR, and therefore (§26) perpendicular to AX, 
and meeting AX at S, AP at T, and PN at N. Then the angle 
iormed by PN and PT is equal to a, the angle formed by AP with 
AX (§ 26). But this angle is the complement of that formed by 
PN and PQ (which tlierefore equals 90° — a), and this, again, is 
(§30) the complement of the angle formed by QP and QN. The 
latter angle, therefore, is equal to a. Hence (§§ 41, 44), 

RP NP . ^ PQ 

sm a = or -? sin p — 



AP QP AQ 

AR QN ^ o AP 

cos a = . or 1 and cos [3 = 



AP QP AQ 

Also the distance RP == SN (§35). 

Now, in Fig. 24 or 25, sin (a + /3) = ^^; 

AQ 

but * SQ = SN + NQ == RP 4- NQ. 

Whence M- = -^ + ^ 



AQ AQ AQ 

M RP RP _ AP . 

Now, ■ = ■ X ^=3- = sin a cos /3 ; 

. AQ AP AQ 

, NQ QN _ PQ 

and ■ ^ = ^ X = cos a sin (3. 

AQ QP AQ 

Substituting these values it appears that 

sin (« + jQ) = sin a cos (3 -\- cos a sin (3, 

In like manner, in Fig. 26, 
sin (a - 



34 FUNCTIONS OF ANGLES. 

-nzr- X X -zzir- =^ sin a cos p — cos a sm iS. 

AP AQ QP AQ 

[It should be noticed that this formula may be obtained 
directly from the other ; for, by substituting ( — (3) for (3 in 

sin (a -\- ft) ^ sin a cos ft -\- cos a sin (3, 
since sin ( — ft) = — sin /S, and cos ( — ft) = cos ft, 

we have sin (a — 13) = sin a cos (3 — cos a sin ft,'] 

98. Cos (a + i3) = sin [90^^ _ (a + ^3)] (§ 86), 

= sin [(90° — a) ~ 13] = sin (90° — a) cos i3 
— cos (90° — a) sin (3 = cos a cos [3 — sin a sin (3. 
Cos (a — i3) = sin [90° — (a — /3)] = sin [(90° — a) 
+ n = sin (90° — a) cos /3 + cos (90° — a) sin i3 
= cos a COS jQ -f- sin a sin j8. 

99. These formulae have now been demonstrated for the case in 
which a and (3 are each less than 90°. If either or each of them is 
numerically greater than 90°, let y and d be two angles each less than 
90°, which differ from a and (3 respectively by some multiple of 
90° by a whole number. Then the difference between a -{- (3 and 
/' + <5 will be either zero or a multiple of 90°, and the same wilt 
be true of the difference between a — (3 and y — d. Now, the 
demonstrations of the two preceding §§ furnish expressions for 
the sine and cosine of ;' + ^ and y — d; and from these, by Table 
II. (§ 94), we may find the like functions of a-\-l3 and a — /3. 
In every *case we shall find that 

sin {a-\- (3) = sin a cos (3 -\- cos a sin |3, 
sin (a — jS) = sin a cos (3 — cos a sin j3, 
cos (a-\- (3) = COS a COS (3 — sin a sin j3, 
cos (« — 13) = cos a COS [3 4" sin a sin (3. 
Hence these formulae are true for all values of a and (3. 
For instance, let a be an angle between — 90° and — 180°, and 
let /3 be an angle between + 270° and -f 3G0\ Then a = ;- — 180° 
and jS = ^ + 270°, where both y and d are between 0° and 90°. 
Hence, 

sin (^a + f^) = sin [(r — 180°) + (^ + 270°)] 

= sin [(r + ^) + 80°] = cos (r -f- O 
= cos y cos d — sin y sin d. 



rUKCTIONS OF ANGLES. 35 

Eestoring a and i3 to the equation, by substituting (180 + a) for 
Y and (jS — 270°) for ^, this expression becomes 
cos (180° + a) cos id — 270°) — sin (180° + a) sin (/3 — 270°) 
= (see § 87) cos (180° + a) cos (270° — (i) + sin (180° 

+ a) sin (270° — (i) 
= (by Table II. ) (— cos a) (— sin /3) + (— sin a) (— cos (i) 
= cos a sin i3 -(- sin a cos [i 
= sin a cos (3 -\- cos a sin /?. 
In the same way, we may find 

cos (a — i3) = cos ( r — "^ — 450°) = sin (r — ^) 
= sin y cos d — cos y sin d 
=: ( — sin a) ( — sin (3) — ( — cos a cos (3) 
= cos a cos i3 + sin a sin i3. 
Similar processes will lead to similar results for any other values 
of a and 13. 

100. The tangent of the sum of two angles is equal to the 
sum of their tangents, divided by 1 minus the product of 
their tangents ; i.e., 

tan a + tan /5 

tan (a + /3) = — ' -' 

^ ' ^ 1 — tan a tan /3 

The tangent of the difference of two angles is equal to the 
difference of their tangents, divided by 1 ;plus the product of 
their tangents ; i.e., 

, _. tan a — tan (3 

tan (a — /3) = ■■ , . ^• 

^ ^ 1 4- ^an a tan (3 

The cotangent of the sum of two angles is equal to the 
product of their cotangents minus 1, divided by the sum of 
their cotangents ; i.e., 

. , _. cot a cot ^ — 1 

cot ( a + /?) = — j —^ — 

^ ' ^ cot a + cot i3 

The cotangent of the difference of two angles is equal to 
the product of their cotangents plus 1, divided by the differ- 
ence of their cotangents taken in the contrary order; i.e., 

^ , „. cot a cot j3 4- 1 

cot (« d^ = 7^ ' 

^ ^ cot i3 — cot a 



36 FUNCTIONS OF ANGLES. 



These formulse are readily deduced from those of § 97 by the 

tan = — ■ and cot = ^- • Thus 
cos sin 

sin (a -\- 13) sin a cos /S -}~ cos a sin j3 



Sin pos 

use of the relations tan = — ■ and cot = — • Thus : 

cos sin 



tan (a + i3) = . , r,^ — 

' cos (a -J- i^) <^0S a COS p Sill a Sin , 

Dividing both terms of this fraction by cos a cos /S, we have 

sin a sin (3 

. , „. cos a •" COS (3 tan a -4- tan (3 

tan (a -j- 13) = — ' ' 



1 — tan a tan [3 











J. - 


COS a 


In 


like 


manner, 










tan 


(«- 


■l3) 


sin 
cos 


(a- 


■0) 

■/3) 










sin a 

cos a 


sin 

cos 


9 
l3 



cosjS 

sin a cos (3 — cos a sin /3 
cos a cos jQ -(- sin a sin 13 

tan « — tan (3 
sin a sin [3 1 -|- tan a tan [3 

cos a COS p 

In deducing the formulae for cotangents, both terms of the frac- 
tion are divided by sin a sin /3, thus: — 

cos (« 4" ^) c^s ^ <^<^s i^ — sin a sin /5 



^u» 


^^^ ' ^^ sin(a + /?) 




COS a ^ COS /3 ^ 

sin a ^^ sin /3 




COS (3 1 cos a 
sin [3 sin a 


301 


^ "^^ Sin (« — f:l) 


\ 


cos a cos /3 

X . ^ "T" •'■ 

Sin a sm p 



sin a 


COS /3 -|- cos a 


sin 


/J 


cot 


a cot (3—- 1 






cot 


a + cot i3 




cos a 


COS /3 -f- sin a 


sin 


/3 


sin a 


cos (3 — cos a 


sin 


/3 


cot 


a cot /? + 1 







cos (Q cos a cot /5 — cot a 

sin i3 sin a 

101. The formulge of §§ 97-100 are collected in the fol- 
lowing table : 

Table III. 
A. sin (a -j- |3) = sin a cos (3 -\- cos a sin /3. 



FUNCTIONS OF ANGLES. 37 

B. sin (a — i^) = sin a cos /3 — cos o. sin /?. 

C. cos (a -(- iS) = cos a cos jS — sin a sin /?. 

D. cos (a — i^) = COS a COS /3 -|- sin a. sin j3. 

_ ^ . , ^- tan a + tan /3 

E. tan (a + |3) = ! — -. 

^ ' ^ 1 — tan a tan |3 

_ ^ , _. tan a — tan 3 

F. tan (a — '^) = f 



G. cot (a + i3) 
H. cot (a — /?) 



-)- tan a tan ,3 

cot a cot (S — 1 
cot a -(- cot /3 

cot a cot /3 -|- 1 



cot /3 — cot a 

102. If, in formulas A, C, E, and G of Table III., we suppose 
j3 == «, we may write at once the following table of functions of 
double angles : — 

sin 2 a = 2 sin a cos a, 

cos 2 a = cos^ a — sin^ a, 

2 tan « 
tan 2 a = 

cot 2 a = 



1 — tan^ a 
cot' a — I 



2 cot a 

Since cos^ a = 1 — sin^ «, and sins a = 1 — cos2 a, 

(§ 48,) we may write the value of cos 2 a in either of the follow- 
ing forms : — 

cos 2 « = 2 C0S2 a — 1, or cos 2 a = 1 — 2 sin''' a. 

Also, since • = tan «, 

cot 6C ' 

cot 2 « = ^ (cot* a — tan a). 
103. Sin 3 a = sin (2 a -\- a) == sin 2 a cos a ~\- cos 2 a sin a = 
2 sin a cos^ a -\- (cos' a — sin' a) sin' a = 3 sin a cos'' a 
— sin^ a. 
If in this value we substitute for cos' a, 1 — sin^ a, we have 

sin 3 a = 3 sin a — 4 sin^ a. 
In like manner we may find that 

cos 3 a = 4 cos^ a — 3 cos a, 
3 tan a — tan'^ a 



tan 3 
cot 3 a = 



1 — 3 tan' a 
COt^ a — 3 cot a 
3 cot' a — 1 



38 PARTICULAR VALUES OF THE FUNCTIONS. 

Sin 4 a = sin (2 X 2) a = 2 sin 2 a cos 2 a 
= 4 sin a cos a (cos^ a — sin^ a). 
In like manner we may find the other functions of 4 a. The 
functions of 5 6c may be found by regarding it as the sum of 4 a 
and «, or of 3 a, and 2 a, etc. 

104. Since cos 2 a = 1 — 2 sin^ a, 

sin^ a = ^ ( 1 — cos 2 a) ; 
and since cos 2 a = 2 cos^ a, — 1, 

COS^ a = ^ ( 1 -f- cos 2 a). 

Since no limitation has been put on the values of a and 2 a, 
save that one is double the other, we may write the above equa- 
tions 

sin^ ^ a = ^ (1 — cos a) and cos'' i- « = ^ (1 + cos a), 
whence 



sin^a = A/^(l — cos a) and cos J-a = ^^(l-[- cos a). 
Also, 

sin 4- a 



tan ^ a 



COS^ a 



cos 



1 — 


COS a 


1 + 


COS a 


11+ 


COS a 



and cot 4- a = i^:::^- 

sin ^ a ^ 1 — COS a 

Other forms of the values of tan ^ a and cot ^ a may be obtained 
from the values of tan 2 a and cot 2 a given in § 102. The signs 
of all these functions of half angles are determinate in every 
case, being fixed by the magnitude of the angles. 

PARTICULAR VALUES OF THE FUNCTIONS. 

105. If, in formulae B, D, F, and H, of § 101, we make /5 = a, 
we shall have sin (a — a) = sin == sin a cos a — sin a cos a =0. 
cos 0° = cos^ a -\- sin^ a = 1, (§ 48,) 

tan 0° = , , ^ , = 0, 
1 + tan^ a 

cot' a + 1 
cot 0° = — = 00. 

By reference to Table II. (§ 94), we may now find the 
values of the functions of 90^, 180% etc. (since 90° = 90° 



PAKTICULAR VALUES OF THE FUNCTIOiSS. 39 

± 0°, etc.) These values are collected in the following 
table, the signs in which are taken from Table I., and indi- 
cate the signs of angles a little less and a little greater than 
the given angle. Thus, the sign of 180° is written d= ; by which 
is meant the sin 180° = 0, and that an angle a little less than 180° 
has a positive sine, while an angle a little greater has a negative 
sine. (In the ambiguous sign zh or =F, the upper sign belongs to 
the function of the angle a little less than the given angle, and the 
lower sign to that of the angle a little greater.) When only one 
sign is written, it is understood that the function has that sign, 
both for the given value of the angle, and for values a little greater 
and a httle less. 

Table IV. 





0° 


90° 


180° 


270° 


360^ 


Sine 


=F 


+ 1 


d= 


— 1 


=F 


tangent 


T 


± GO 


=F 


=b ^ 


=F 


cosine 


+ 1 


z±z 


— 1 


=F 


+ 1 


cotangent 


=F <^ 


± 


=F <x 


d= 


=F 00 



This table, like Tables I. and II. may be extended indefinitely 
in either direction, and such an extension will be simply a repeti- 
tion of four different columns. 

106. From § 96, we know that the angles included in table IV. 
are the only angles having the given functions; i.e., that no other 
angles can have sines or tangents equal to 0, except 0°, 180°, 
360°, and these increased or diminished by the multiples of 180° ; 
no cosine can be equal to + 1 except those of 0° and multiples of 
360°, positive or negative, etc. Moreover, the values given in 
this table are the limiting values of the function ; that is, they are, 
algebraically or numerically, the greatest and least values possible 
for the functions. It is evident that no finite quantity can be 
greater than oo, or numerically less than 0, or algebraically less 
than — CO, and these values embrace all given in the table, except 
those of the sines of 90°, 270°, etc., and the cosines of 0°, 180°, 
etc., which are equal to + 1 ^i* — 1- Hence it is to be shown 
only that no sine or cosine can be numerically greater than 1. 
This readily appears from the equation sin'^ a-|- cos^ a z= 1 (§ 48), 
where sin^ a and cos^ a must be positive, whether sin a and cos a 



40 PARTICULAK VALUES OF THE FUNCTIONS. 

are so or not, therefore neither of them can exceed 1 ; and hence 
neither sin a or cos a can numerically exceed 1, whatever the 
value of a. 

107. Hence the sines of angles between 0° and 90° are inter- 
mediate in value between and -f- 1, their cosines are between 
-}- 1 and 0, their tangents between and -f- co, etc. The sines 
of angles between 90° and 180° are intermediate in value between 
-|- 1 and 0, etc. If 6 be an angle supposed to increase continu- 
ously in value, commencing at 0°. sin 6 will be when d is 0, and 
will increase gradually till 6 = 90°, when it will be -j- 1, then it 
will diminish until = 180°, when it will again equal ; as ^ 
passes from 180° to 270°, sin d will become negative and will con- 
tinue to diminish algebraically, but increase numerically, till, when 
6 = 270°, sin = — 1. From that point it will increase alge- 
braically, but decrease numerically, till d = 360°, when sin 
again equals 0, and thence the same changes are repeated in the 
same order. 

Tan 0, when = 0, is also equal to 0, and, like sin d, increases 
as d approaches 90°, but more rapidly, so that when 6 = 90°, 
tan 6 = CO. Tan 6 has been positive up to this point, but here 
changes sign from -f- <^ to — oo, and as d increases in value 
toward 180°, continues to decrease numerically, but increase alge- 
braically to 0, which value it reaches when 6 == 180°. In passing 
through this value it changes sign, and continues to increase till 
6 = 270°, when it is again infinite, and again changes sign, and 
from this point decreases numerically till it becomes equal to 0^ 
when (9 = 360°. 

[Let the student trace the corresponding changes in the value of cos Q and cot 
0, as well as the values of all the functions when Q is a negative angle, increas- 
ing numerically; and let him notice that the functions of angles change their 
sign when, and only when, these functions pass through the values of and go. 
It is an established principle in Geometry as well as Algebra, that no varying 
function can change sign, except in passing through one of these values. The 
converse proposition, however, that a function in passing through or oo 
always changes sign, though true of these four functions of angles, as we have 
seen, is not a principle of general application.] 

108. Besides the angles which are multiples of 90°, and whose 
functions have the limiting values, there are other angles whose 
functions are of frequent use ; particularly 45°, 30°, and 60°. 



PAETICULAR VALUES OF THE FUNCTIONS. 41 

If a = 45% a = 90° — a, .*. sin a = sin (90° — a) = cos a. 

But sin^ a + cos^ a= 1 (§ 48) .*. 2sin^a=l, 

or sin a r= cos a^=i yj ^ = ^ \J 2. 

sin a _ cos a 

Now, as tan a =: ■ and cot a == r' 

cos a sin a 

when sin a = cos a, eacli of these is equal to 1. 

.-. sin45° = iV2, cos45° = iV2, 

tan 45° = 1, cot 45° ==1. 

If a = 30°, 90° — a = 2 a, or sin (90° — a) == sin 2 a, 

cos a zzz 2 sin a cos a ; whence, sin a= J- ; 



cos a = \/ 1 — sin^ ar=z^^z=^A^/3; 

cos a .-5- 

cot a = — = V d ; 

sm a 

tan a = — : — = a/ i = 1 v^ 3. 
cot a ^ "^ " ^ 

sin 30° = ^; cos30° = ^/v/T; 

tan 30° r=r i V 37 cot 30° = V 3. 

From the cosine of 30° we may find the sine and cosine of 15° 
by the formulae of § 104, and from these the tangent and cotan- 
gent. To find the functions of 60°, we may employ the formulae 
of § 102 ; or, more simply, since 60° is the complement of 30°, 
we may write sin 60° = cos 30°, cos 60° = sin 30°, etc. 

sin 60° r=r J- y^ cos 60° = ^ ; 

tan 60° = V^ cot 60° = i V^ 

109. In a similar manner we may find the functions of 18°. 
If a = 18°, we have 2 a = 90° — 3 a. 

Hence (§§ 102, 103), 

2 sin a cos a = 4 cos^ a — 3 cos a, 
or 2 sin a = 4 cos^ a — 3. 

Substituting for cos^ a its value, 1 — sin^ a, this becomes 

2 sin a = 4 — 4 sin^ a, — 3, 
or 4 sin^ a + 2 sin a = 1, 

whence, by the solution of a quadratic equation, we have 
sin a = i (\/ o — 1). 



42 PARTICULAR VALUES OF THE FUNCTIONS. 

Only one of the roots of this equation (the positive one) can be 
taken as the sine of 18°, since the angle is less than 90°. For a 
similar reason, all the functions in the preceding section are 
positive. 

110. To compute the sine and cosine of V. — From the formulae 
of §§ 102 and 103 we may obtain the equations: 

sin 2 a 



2 sin a 
sin 3 a 

3 sin a 
sin 4 a 



= cos a = \/ \ 



sin a. 



etc. 



In all these values it is to be noticed that when a is a very small 
angle, and, consequently, sin a is a very small fraction, sin^ a will 
be much smaller, and the second member of each equation will be 
very nearly equal to 1. Hence, when a is a very small angle, 

sin 2 a ^ sin 3 a ^ sin 4 a ^ 

= 2, -. == 3, -: = 4, etc., very nearly; 

sm a sni a sin a "^ ^ 

that is, the sines of small angles are very nearly in the same 
ratio as the angles themselves. Hence, if we can find the sine 
of some angle not larger than 1', we may thence find the sine 
of 1', with sufficient exactness for all practical purposes, by mul- 
tiplying that sine by the ratio of 1' to the corresponding angle. 
In order to find the sine of this small angle, we may first calculate 
the cosine of 3° from the formula cos (a — i^) = cos a cos (3 -|- 
sin a sin jS, since 3° is the difference of 18° and 15°, whose func- 
tions are known. Then, by successive applications of the formula 
of § 104, the cosine of the half, the quarter, etc., of 3° may be 
obtained, to as small an angle as we wish, and from the cosine of 
this angle we may find the sine of an angle half as large. In this 
way, having computed from the formula of § 108 the values 

sin 15° = .25881904510252, 
cos 15° =.96592582627196, 
and from § 109, 

sin 18° = .30901699437495, 
cos 18° = .95105651629515, 



PARTICULAR VALUES OF THE FUNCTIONS. 43 

we obtain cos 3° = .998629534738030; 

and thence, successively, 

cos 1° 30'= .99965732497149, 

cos 45' = .99991432757299, 

cos ^-' = .99997858166387, 

cos^'= .99999464540163, 

cos-V-'== .99999866134951, 

cos ff ' = .99999966533732, 
nnri cos y-' = .99999991633433. 

From cos ^^' we obtain 

sin ff' = .000409061534, 
and from cos If , sin ||-' = .000204530772. 

We now observe that the half of sin ^' differs from sin f|-' by 
not more than 5 in the twelfth place of decimals. Hence we 
infer that if we take ff of sin |-|-', it will be the value of the sine 
of ft of fl', that is, of 1', correct to at least ten or eleven places 
of decimals. The value to ten places is, 

sin 1' = .0002908882. 
The cosine of 1' may now be found from the formula cos 1' 
= \/ 1 — sin' 1'. Its value is, cos 1' = .9999999577. 

111. To compute a table of siries^ tangents^ etc. — By adding and 
subtracting the first four formulae of § 101 we obtain the following : 
sin (a -{- 13) -\- sin (a — j3) = 2 sin a cos /9, (1) 

sin (a -\- (3) — sin (a — (3) == 2 cos a sin /?, (2) 

cos (a + /3) -f- cos (a — j3) = 2 cos a cos /?, (3) 

cos (fic — j3) — cos (a -|- i3) = 2 sin a sin /?. (4) 

If in (1) and (3) we make (3 constantly equal to 1', and let a equal 
successively 1', 2', 3', etc., we shall obtain, by transposition, 

sin (1' + 1') = sin 2' = 2 sin 1' cos 1' — sin 0' ^ when a = V 
cos 2' = 2 cos 1' cos 1' — cos 0' \ and [3 = 1'. 

sin 3' = 2 sin 2' cos 1' — sin 1' ) when a = 2' 

cos 3' = 2 cos 2' cos 1' — cos 1' ) and 13=1'. 

etc., etc., etc., etc., 

and by these formulae we may calculate successively the sine and 

cosine of 2', 3', 4', etc., up to 30°. We may now avoid the labor 



44 PARTICULAP. VALUES OF THE FUNCTIONS. 

of further multiplications by the use of the formulae (1) and (4) 
of this section, in which we make a constantly equal to 30°, and 
let /? equal 1', 2', 3', etc., snccessivel3\ 
Rememberinof that sin 30^ = ^. we have, 

sin 30° 1' = cos 1' — sin 29° 59' > , „.„ , ^ ., 

oAo-,, ono^nf • 1/ > whcu a = 30 and A? = 1'. 

cos 30 1' = cos 29 o9' — sni 1' 5 

sin 30° 2' = cos 2' — sin 29° 58' ) , o^o -. o ^, 

or^o nf onocof • of > whcu « = 30 aud p = 2'. 

cos 30 2' = cos 29 58' — sui 2' > 

etc., etc., etc. 

We may thus obtain by subtraction the sines and cosines of 
angles up to 45°. From this point, by the use of the formulas 
sin a = cos (90° = d) and cos a = sin (90° — a), we may write 
at once the sines and cosines of angles from 45° to 90°. It will not 
be necessary to extend the table beyond 90°, since by Table II., 
§ 94, we may find the functions of any angles whatever, from 
those of angles between 0° and 90°. 

A table of tangents may now be formed by dividing each sine by 

,. . . , sin a . T 1 , 1 . , 

its corresponding cosine, since tan a = ? and by takmoj each 

^ * cos a JO 

tangent thus found as the cotangent of the complementary angle, 
a table of cotangents is obtained. 

Tables of Natural Sines and Tangents are given on pages 45^and 46, and 
Table of Reciprocals on pages 47 and 48. 



NAT LEAL SINES. 



45 



Table Y. Natural Sines. 





0' 


10' 


20 


30' 


40 


50' 


D 




0' 10' 


20' 


30' 


40 


50 


D 


0° 


0000 


0029 


0058 


0087 


0116 


0145 


2.9 


45° 


7071:7092 


7112 


7132 


7153 


7173 


2.0 


1 


0175 


0204 0233 


0262 


0291 


0320 


2.9 


46 


7193:7214 


7234 


7254 


7274 


7294 


2.0 


2 


0349 


0378,0407 


0436 


0465 


0494 


2.9 


47 


7314i7333 


7353 


7373 


7392 


7412 


2.0 


3 


0523 


0552!058li0610 


0640 


0669 


2.9 


48 


743117451 


7470 


7490 


7509 


7528 


1.9 


4 


0698 


07271075610785 


0814 


0843 


2.9 


49 


7547|7566 


7585 


7604 


7623 


7642 


1.9 


5 


0872 


09011092910958 


0987 


1016 


2.9 


50 


7660:7679 


7698 


7716 


7735 


7753 


1.9 


6 


1045 


1074 110311132 


1161 


1190 


2.9 


51 


777li7790 


7808 


7826 


7844 


7862 


1.8 


7 


1219 


1248 


1276 


1305 


1334 


1363 


2.9 


52 


7880 7898 


7916 


7934 


7951 


7969 


1.8 


8 


1392 


1421 


1449 


1478 


1507 


1536 


2.9 


53 


7986 8004 


8021 


8039 


8056 


8073 


1.7 


9 


1564 


1593 


1622 


1650 


1679 


1708 


2.9 


54 


8090 8107 


8124 


8141 


8158 


8175 


1.7 


10 


1736 


1765 


1794 


1822 


1851 


1880 


29 


55 


8192 8208 


8225 


8241 


8258 


8274 


1.6 


11 


1908 


1937 


1965 


1994 


2022 


2051 


2.9 


56 


8290 8307 


8323 


8339 


8355 


8371 


1.6 


12 


2079 


2108 2136 


2164 


2193 


2221 


2.8 


57 


8387 8403 


8418 


8434 


8450 


8465 


1.6 


13 


2250 


2278!2306 


2334 


2363 


2391 


2.8 


58 


848018496 


8511 


8526 


8542 


8557 


1.5 


14 


2419 


2447:2476 '2504 


2532 


2560 


2.8 


59 


8572 8587 


8601 


8616 


8631 


8646 


1.5 ' 


15 


2588 


261612644 2672 


2700 


2728 


2.8 


60 


8660 


8675 


8689 


8704 


8718 


8732 


1.4 


16 


2756 


2784:2812 2840 


2868 


2896 


2.8 


61 


8746 


8760 


8774 


8788 


8802 


8816 


1.4 


17 


2.J24 


2952 2979 3007 


3035 


3062 


2.8 


62 


8829 


8843 


8857 


8870 


8883 


8897 


1.8 


18 


3090 


3118 


314513173 


3201 


3228 


2.8 


63 


8910 


8923 


8936 


8949 


8962 


8975 


1.3 


19 


3256 


3283 331113338 


3365 


3393 


2.7 


64 


8988 


9001 


9013 


9026 


9038 


9051 


1.3 


20 


3420 


3448|3475|3502 


3529 


3557 


2.7 


65 


9063 


9075 


9088 


9100 


9112 9124 


1.2 


21 


3584 


36113638:3665 


3692 


3719 


2.7 


66 


9135 


9147 


9159 


9171 


9182 9194 


1.2 


22 


3746 


3773i3800|3827 


3854 


3881 


2.7 


67 


9205 


9216 


9228 


9239 


9250 9261 


1.1 


23 


3907 


3934!396l'3987 


4014 


4041 


2.7 


68 


9272 


9283 


9293 


9304 


9315 9325 


1.1 


24 


4037 


4094i4120 4147 


4173 


4200 


2.6 


69 


933619346 


9356 


9367 


9377 


9387 


1.0 


25 


4226 


42531427914305 


4331 


4358 


2.6 


70 


9397 9407 


9417 


9426 


9436 


9446 


1.0 


26 


4384 


4410 


4436 


4462 


4488 


4514 


2.6 


71 


9455 


9465 


9474 


9483 


9492 


9502 


0.9 


27 


4540 


4566 


4592 


4617 


4643 


4669 


2.6 


72 


9511 


9520 


9528 


9537 


9646 


9554 


0.9 


28 


4695 


4720 


4746 


4772 


4797 


4823 


2.6 


73 


9563 


9572 


9580 


9588 


9596 


9605 


0.8 


29 


4848 


4874 


4899 


4924 


4950 


4975 


2.5 


74 


9613 


9621 


9628 


9636 


9644 


9652 


0.8 


30 


5000 


5025 


505015075 


5100 


5125 


2.5 


75 


9659 


9667 


9674 


9681 


9689 


9696 


0.7 


31 


5150 


5175 


52005225 
5348 '5373 


5250 


5275 


2.5 


76 


9703 


9710 


9717 


9724 


9730|9737 


0.7 


32 


5299 


5324 


5398 


5422 


2.5 


77 


9744 


9750 


9757 


9763 


976919775 


0.6 


33 


5446 


5471 


5495 


5519 


5544 


5568 


2.4 


78 


9781 


9787 


9793 


9799 


9805! 9811 


0.6 


34 


5592 


5616 


5640 


5664 


5688 


5712 


2.4 


79 


9816 


9822 


9827 


9833 


9838,9843 


0.5 


35 


5736 


5760 


5783 


5807 


5831 


5854 


2.4 


80 


9848 


9853 


9858 


9863 


9868 


9872 


0.5 


36 


5878 


5901 


5925 


5948 


5972 


5995 


2.3 


81 


9877 


9881 


9886 


9890 


9894 


9899 


0.4 


37 


6018 


6041 


6065 


6088 


6111 


6134 


2.3 


82 


9903 


9907 


9911 


9914 


9918 


9922 


0.4 


38 


6157 


6180 


6202 


6225 


6248 


6271 


2.3 


83 


9925 


9929 


9932 


9936 


9939 


9942 


0.3 


39 


6293 


6316 


6338 


6361 


6383 


6406 


2.2 


84 


9945 


9948 


9951 


9954 


9957 


9959 


0.3 


40 


6428 


6450 


6472 


6494 


6517 


6539 


2.2 


85 


9962 


9964 


9967 


9969 


9971 


9974 


0.2 


41 


6561 


6583 


6604 


6626 


6648 


6670 


2.2 


86 


9976 


9978 


9980 


9981 


9983 


9985 


0.2 


42 


6691 


6713 


6734 


6756 


6777 


6799 


2.1 


87 


9986 


9988 


9989 


9990 


9992 9993 


0.1 


43 


■6820 


6841 


6862 


6884 


6905 


3926 


2.1 


88 


9994 


9995 


9996 


9997 


9997 


9998 


0.1 


44 


6947 


6967 


6988 


7009 


7030 


7050 


2.1 


89 


9998 


9999 


9999 


unity 


unity 


uuity 


0.0 




0' 


10 


20' 


30' 


40' 


50' 






0' 1 10' 


20' 


30' 


40 


50' 





46 



NATURAL TANGENTS, 



Table VI. Natural Tangents. 





0' 

0000 


10' 
0029 


20' 
0058 


30' 


40' 


50' 
0145 


D 

2.9 




0' 


10' 


20' 


30 


40' 


50' 


D 


0° 


0087'0116 


45° 


1.000 


1.006 


1.012 


1.018 


1.024 


1.030 


0.6 


1 


0175,0204 


0233 


0262,0291 


0320 


2.9 


46 


1.036 


1.042 


1.048 


1.054 


1.060 


1.066 


0.6 


2 


03490378 


0407 


0437,0466 


0495 


2.9 


47 


1.072 


1.079 


1.085 


1.091 


1.098 


1.104 


0.6 


3 


05240553 


0582 


061210641 


0670 


2.9 


48 


1.111 


1.117 


1.124 


1.130 


1.137 


1.144 


0.7 


4 


03990729 


0758 


07870816 


0846 


2.9 


49 


1.150 


1.157 


1.1H4 


1.171 


1.178 


1.185i 0.7 


6 


0875 


0904 


0934 


0963,0992 


1022 


2.9 


50 


1.192 


1.199 


1.203 


1.213 


1.220 


1.228' 0.7 


6 


1051 


1080 


1110 


1139 1169 


1198 


2.9 


51 


1.235 


1.242 


1.250 


1.257 


1.265 


1.272 0.8 


7 


1228 


1257 


1287 


1317 


1346 


1376 


3.0 


52 


1.280 


1.288 


1.295 


1.303 


1.311 


1.319| 0.8 


8 


1405 
1584 


1435 
1614 


1465 
1644 


1495 
1673 


1524 
1703 


1554 
1733 


3.0 
3.0 


53 


1.327 


1.335 


1.343 


1.351 


1.360 


1.368 


0.8 


9 


54 


1.376 


1.885 


1.393 


1.402 


1.411 


1.419 


0.9 


10 


1763 


1793 


1823 


1853 


1883 


1914 


3.0 


55 


1.428 


1.437 


1.446 


1.455 


1.464|1.473 


0.9 


11 


1944 


1974 


2004 


2035'2085 


2095 


3.0 


56 


1.483 


1.492 


1.501 


1.511 


1.5201.530 


1.0 


12 


2126 


2156 


2186 


2217J2247 


2278 


3.1 


57 


1.540 


1.550 


1.560 


1.570 


1.580 1.590 


1.0 


13 


2309 2339 


2370 


2401 2432 


2462 


3.1 


58 


1.600 


1.611 


1.621 


1.632 


1.643 1.653 


1.1 


14 


2493:2524 


2555 


2586 2617 


2648 


3.1 


59 


1.664 


1.675 


1.686 


1.698 


1.709 1.720 1.1 


15 


26792711 


2742'2773 2805 


2836 


3.1 


60 


1.732 


1.744 


1.756 


1.767 


1.780 1.792i 1.2 


16 


286712899 


2931 


2962 2994 


3026 


3.2 


61 


1.804 


1.816 


1.829 


1.842 


1.855 


1.868 1.3 


17 


3057 3089 
3249'3281 


3121 
3314 


3153,3185 
3346 3378 


3217 
3411 


3.2 
3.2 


62 
63 


1.881 


1.894 


1.907 


1.921 


1.935 


1.949| 1.4 

1 


18 


1.963 


1.977 


1.991 


2.003 


2.020 


2.035 1.5 


19 


3443 '3476 


3508 


35413574 


3607 


3.3 


64 


2.050 


2.066 


2.081 


2.097 


2.112 


2.128 


1.6 


20 


3640 3673 


3703 


373913772 


3805 


3.3 


65 


2.145 


2.161 


2.177 


2.194 


2.211 


2.229 


1.7 


21 


3839 3H72 


3906 


3939 3973 


4003 


3.4 


66 


2.246 


2.264 


2.282 


2.300 


2.318 


2.337 


1.8 


22 


4040 4074 


4108 


4142,4176 


42 


3.4 


67 


2.356 


2.375 


2.394 


2.414 


2.434 


2.455 


2.0 


23 


4245 4279 


4314 


4348 '4383 


4417 


3.5 


68 


2.475 


2.498 


2.517 


2.539 


2.560 


2.583 


2.2 


24 


4452 


4487 


4522 


4557J4592 


4628 


3.5 


69 


2.605 


2.628 


2.651 


2.675 


2.699 


2.723 


2.4 


25 


4663 


4699 


4734 


4770 4806 


4841 


3.6 


70 


2.747 


2.773 


2.798 


2.824 


2.850 


2.877 


2.6 


26 


4877 


4913 


4950 


4986 5022 


5059 


3.6 


71 


2.904 


2.932 


2.960 


2.989 


3.018 


3.047 


2.9 


27 


5095 


5132 


5169 


5203 5243 


5280 


3.7 


72 


3.078 


3.108 


3.140 


3.172 


3.204 


3.237 


8,2 


28 


5317 


535-4 


5392 


5430 5467 


5505 


3.8 


73 


3.271 


3.305 


3.340 


3.376 


3.412 


3.450 




29 


5543 


5581 


5619 


5658 5696 


5735 


3.8 


74 


3.487 


3.526 


3.566 


3.603 


3.647 


3.689 




30 


5773 


5812 


5851 


5890 5930 


5969 


3.9 


75 


3.732 


3.776 


3.821 


3.867 


8.914 


3.962 




31 


6009 


6048 


6088 


6128,6168 


6208 


4.0 


76 


4.011 


4.031 


4.113 


4.165 


4.219 


4.275 




32 


6249 


6289 


6330 


637116412 


6453 


4.1 


77 


4.331 


4.390 


4.449 


4.511 


4.574 


4.638 




33 


6494 


6536 


6577 


6619,6661 


6703 


4.2 


78 


4.705 


4.773 


4.843 


4.915 


4.989 


5.036 




34 


6745 


6787 


6830 


68736916 


6959 


4.3 


79 


5.145 


5.226 


5.309 


5.396 


5.485 


5.576 




35 


7002 

7265 


7046 
7310 


7089 
7355 


71337177 
7400 7445 


7221 
7490 


4.4 

4.5 


80 

81 


5.671 


5.769 


5.871 


5.976 


6.084 


6.197 




36 


6.314 


6.435 


6.561 


6.691 


6.827 


6.968 




37 


7536 


7581 


7627 


76737720 


7766 


4.6 


[82 


7.115 


7.269 


7.429 7.596 


7.770 


7.953 




38 


7813 


7860 


7907 


7954 8002 


8050 


4.7 


83 


8.144 


8.345 


8.55618.777 


9.010 


9.255 




39 


8098 


8146 


8195 


8243 


8292 


8342 


4.9 


84 


9.514 


9.788 


10.08ilO.39 


10.71 


11.06 




40 


8391 


8441 


8491 


8541 


8591 


8642 


5.0 


85 


11.43 


11.83 


12.25ll2.71 


13.20 


13.73 




41 


8693 


8744 


8796 


8847 


8899 


8952 


5.2 


86 


14.30 


14.92 


15.60116.35 


17.17 


18.07 




42 


9004 


9057 


9110 


9163 


9217 


9271 


5.4 


87 


19.08 


20.21 


21.47 


22.90124.54 


23.43 




43 


9325 


9380 


9435 


9490 


9545 


960! 


55 


88 


28.64 


31.24 


34.37 


38.19142.96 


49.10 




44 


9657 



9713 
10 


9770 

20' 


9827 
30' 


9884 
40' 


9942 
50' 


5.7 


89 


57.29 


68 75 


85.94 


114.6 


171.9 


343.8 
50' 








0' 


10' 


20' 


30' 


40' 











TABLE 


VII. 


llECirKOCALS. 






47 







1 


2 


3 


4 


5 
19524 


6 
9434 


7 
9346 


8 
9259 


9 


D 


10 


10000 


9901 


9804 


9709 


9615 


9174 


91.6 


11 


9091 


9009 


9929 


8850 


8772 


8696 


8621 


8547 


8475 


8403 


76.3 


12 


8333 


8264 


8197 


8130 


8065 


8000 


7937 


7874 


7813 


7752 


64.5 


13 


7692 


7634 


7576 


7519 


7463 


7407 


7353 


7299 


7246 


7194 


55.2 


14 


7143 


7092 


7042 


6993 


6944 


6897 


6849 


6803 


6757 


6711 


47.9 


15 


6667 


6623 


6599 


6536 


6494 


6452 


6410 


6369 


6329 


6289 


41.9 


16 


6250 


6211 


6173 


6135 


6098 


6061 


6024 


5988 


5952 


5917 


37,0 


17 


5882 


5848 


5814 


5780 


5747 


5714 


5682 


5650 


5618 


5587 


32.8 


18 


5556 


5525 


5495 


5464 


5435 


5405 


5376 


5348 


5319 


5291 


29.3 


19 


5263 


5236 


5208 


5181 


5155 


5128 


5102 


5076 


5051 


5025 


26.4 


20 


5000 


4975 


4950 


4926 


4902 


4878 


4854 


4831 


4808 


4785 


23.9 


21 


4762 


4739 


4717 


4695 


4673 


4651 


4630 


4608 


4587 


4566 


21.7 


22 


4545 


4525 


4505 


4484 


4464 


4444 


4425 


4405 


4386 


4367 


19.8 


23 


4348 


4329 


4310 


4292 


4274 


4255 


4237 


4219 


4202 


4184 


18.2 


24 


4167 


4149 


4132 


4115 


4098 


4082 


4065 


4049 


4032 


4016 


16.7 


25 


4000 


3984 


3968 


3953 


3937 


3922 


3906 


3891 


3876 


3861 


15.4 


26 


3846 


3831 


3817 


3802 


3788 


3774 


3759 


3745 


3731 


3717 


14.3 


27 


3704 


3690 


3676 


3663 


3650 


3636 


3623 


3610 


3597 


3584 


13.3 


28 


3571 


3559 


3546 


3534 


3521 


3509 


3497 


3484 


3472 


3460 


12.3 


29 


3448 


3436 


3425 


3413 


3401 


3390 


3378 


3367 


3356 


3344 


11.5 


30 


3333 


3322 


3311 


3301 


3289 


3279 


3268 


3257 


3247 


3236 


10.8 


31 


3226 


3215 


3205 


3195 


3185 


3175 


3165 


3155 


3145 


3135 


10.1 


32 


3125 


3115 


3106 


3096 


3086 


3077 


3067 


3058 


3049 


3040 


9.5 


33 


3030 


3021 


3012 


3003 


2994 


2985 


2976 


2967 


2959 


2950 


8.9 


34 


2941 


2933 


2924 


2915 


2907 


2899 


2890 


2882 


2874 


2865 


8.4 


35 


2857 


2849 


2841 


2833 


2825 


2817 


2809 


2801 


2793 


2786 


8.0 


36 


2778 


2770 


2762 


2755 


2747 


2740 


2732 


2725 


2717 


2710 


7.5 


37 


2703 


2695 


2688 


2681 


2674 


2667 


2660 


2653 


2646 


2639 


7.1 


38 


2632 


2625 


2618 


2611 


2604 


2597 


2591 


2584 


2577 


2571 


6.& 


39 


2564 


2558 


2551 


2545 


2538 


2532 


2525 


2519 


2513 


2506 


6.4 


40 


2500 


2494 


2488 


2481 


2475 


2469 


2463 


2457 


2451 


2445 


6.1 


41 


2439 


2433 


2427 


2421 


2415 


2410 


2404 


2398 


2392 


2387 


5.8 


42 


2381 


2375 


2370 


2364 


2358 


2353 


2347 


2342 


2336 


2331 


5.6 


43 


2326 


2320 


2315 


2309 


2304 


2299 


2294 


2288 


2283 


2278 


5.3 


44 


2273 


2268 


2262 


2257 


2252 


2247 


2242 


2237 


2232 


2227 


5.1 


45 


2222 


2217 


2212 


2208 


2203 


2198 


2193 


2188 


2183 


2179 


4.8. 


46 


2174 


2169 


2165 


2160 


2155 


2151 


2146 


2141 


2137 


2132 


4.& 


47 


2128 


2123 


2119 


2114 


2110 


2105 


2101 


2096 


2092 


2088 


4.4 


48 


2083 


2079 


2075 


2070 


2066 


2062 


2058 


2053 


2049 


2045 


4.3 


49 


2041 


2037 


2033 


2028 


2024 


2020 


2016 


2012 


2008 


2004 


4.1 


50 


2000 


1996 


1992 


1988 


1984 


1980 


1976 


1972 


1969 


1965 


3.9 


51 


19608 


9569 


9531 


9493 


9455 


9417 


9380 


9342 


9305 


9268 


37.8 


52 


19231 


9194 


9157 


9120 


9084 


9048 


9011 


8975 


8939 


8904 


36.4 


53 


18868 


8832 


8797 


8762 


8727 


8692 


8657 


8622 


8587 


8553 


35.0 


54 


18519 8484 


8450 


8416 


8382 


8349 


8315 


8282 


8248 


8215 


33.7 


55 


1818218149 


8116 8083'805l| 


8018 


7986 


7953 


7921 7889 


32.5 



48 



TABLE VII. — continued. 







56 
57 

58 
59 
60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 
76 
77 
78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 
90 
91 
92 
93 
94 
95 
96 
97 
98 
99 
100 
101 



17857 
17544 
17241 
16949 
16667 
16393 
16129 
15873 
15625 
15385 
15152 
14925 
14706 
14493 
14296 
14085 
13889 



1 



7825 
7513 
7212 
6920 
6639 
6367 
6103 
5848 
5600 
5361 
5129 
4903 
4684 
4472 
4265 
4065 
3870 



13699 3680 
13514!3495 



13333 
13158 
12987 
12821 
12658 



3316 
3141 
2970 
2804 
2642 



125002484 
12346 2330 
12195 2180 
12048 2034 
11905|l891 



11765 
11628 
11494 
11364 
11236 
11111 



1751 
1614 
1481 
1351 
1223 
1099 



10989 j0977 
10870 0858 
10753^0741 
10638 0627 
10526 0515 



10417 
10309 
10204 
10101 
10000 
99010 



0406 
0299 
0194 
0091 
9900 
8912 



7794 

7482 

7182 

6892 

6611 

6340 

6077 

5823 

5576 

5337 

5106 

4881 

4663 

4451 

4245 

4045 

3850 

3661 

3477 

3298 

3123 

2953 

2788 

2626 

2469 

2315 

2165 

2019 

1876 

1737 

1601 

1468 

1338 

1211 

1086 

0965; 

0846 

0730 

0616 

0504 

0395' 

0288 

0183 

0081 

9800 

8814 



7762 7731 
7452 7422 
7153 7123 
6863:6835 
6584 6556 
6313 6287 
6051 6026 
5798 5773 
5552 5528 
5314|5291 
5083 5060 
485914837 
464l|4620 
4430,4409 
4225 4205 
4025:4006 
3831:3812 
3643:3624 
345913441 
328013263 
3106 3089 



2937 
2771 



2920 
2755 



2610 2594 
245312438 
2300!2285 
21512136 



2005 
1862 
1723 
1587 
1455 
1325 
1198 
1074 
0953 
0834 
0718 



1990 
1848 
1710 
1574 
1442 
1312 
1186 
1062 
0941 
0823 
0707 



0604|0593 



0493!0482 
0384;0373 
027710267 



0173 
0070 
9701 

8717 



0163 
0060 
9602 
8619 



7699 
7391 
7094 
6807 
6529 
6260 
6000 
5748 
5504 
5267 
5038 
4815 
4599 
4388 
4184 
3986 
3793 
3605 
3423 
3245 
3072 
2903 
2739 
2579 
2422 
2270 
2121 
1976 
1834 
1696 
1561 
1429 
1299 
1173 
105Q 
0929 
0811 
0695 
0582 
0471 
0363 
0256 
0152 
0050 
9502 
8522 



7668 
7361 
7065 
6779 
6502 
6234 
5974 
5723 
5480 
5244 
5015 
4793 
4577 
4368 
4164 
3966 
3774 
3587 
3405 
3228 
3055 
2887 
2733 
2563 
2407 
2255 
2107 
1962 
1820 
1682 
1547 
1416 
1287 
1161 
1038 
0917 
0799 
0684 
0571 
0460 
0352 
0246 
0142 
0040 
9404 
8425 



7637 
7331 
7036 
6750 
6474 
6207 
5949 
5699 
5456 
5221 
4993 
4771 
4556 
4347 
4144 
3947 
3755 
3569 
3387 
3221 
3038 
2870 
2706 
2547 
2392 
2240 
2092 
1947 
1806 
1669 
1534 
1403 
1274 
1148 
1025 
0905 
0787 
0672 
0560 
0449 
0341 
0235 
0132 
0030 
9305 
8328 



7606 
7301 
7007 
6722 
6447 
6181 
5924 
5674 
5432 
5198 
4970 
4749 
4535 
4327 
4124 
3928 
3736 
3550 
3369 
3193 
3021 
2853 
2690 
2531 
2376 
2225 
2077 
1933 
1792 
1655 
1521 
1390 
1261 
1136 
1013 
0893 
0776 
0661 
0449 
0438 
0331 
0225 
0121 
0020 
9206 
8232 



9 



7575 
7271 
6978 
6694 
6420 
6155 
5898 
5649 
5408 
5175 
4948 
4728 
4514 
4306 
4104 
3908 
3717 
3532 
3351 
3175 
3004 
2837 
2674 
2516 
2361 
2210 
2063 
1919 
1779 
1641 
1507 
1377 
1249 
1123 
1001 
0881 
0764 
0650 
0537 
0428 
0320 
0214 
0111 
0010 
9108 
8135 



DIRECTIONS FOR USING TABLES. 49 

Directions for Using the foregoing Tables. 

A. To find the sine, cosine, tangent, or cotangent of any given 
ano-jo. 

Find from Table II., § 94, an angle between 0° and 90° whose 
sine or tangent will be equal to the required function. If this angle 
is expressed in degrees, minutes, and seconds, divide the number 
of seconds by six and annex the quotient to the minutes as tenths 
of a minute ; thus: 61° 23' 44" = 61°23'.7 + . Find the degrees 
at the left and the tens' figure in the minutes at the top of the 
table; thus: sin 61° 20' = 8774 or tan 61° 20' = 1.829. Then 
multipl}" the units and tenths of minutes by the nu>nber in column 
D opposite the degree; thus, in the table of sines: 1.4 X 3.7 
= 5.2 ; or in the table of tangents, 1.3 X 3. 7 = 4. 8. The nearest 
whole number to the result is the correction, which is to be added 
to the sine or tangent already found. Thus, in each of the above 
cases the correction is 5, whence sin 61° 23'. 7 = .8774 -j- 5 
= .8779 ; and tan 61° 23'.7 == 1.829 + 5 = 1.834. [The decimal 
point is to be prefixed to all sines taken from the table, also to the 
tangents of all angles less than 45°.] 

B. To find an angle whose sine or tangent is given. 

Take from the table the number next less than the given num- 
ber, noting the number of degrees at the left and of minutes at 
the top. Divide the difference between the given number and the 
number taken from the table, by the divisor in the column D at the 
riglit in the same horizontal line. The quotient is minutes and 
decimals of a minute, which are to be added to the degrees and 
minutes already found. 

Thus, to find sin -.6000. The next less sine in the table is 
.5995, which belongs to 36° 50'. The difference is 5, which is to 
be divided by 2.3 from column D. 5^2.3 = 2.2 and 36° 50' 
+ 2'.2 = 36° 52'.2 the required angle. 

C. To find an angle whose cosine or cotangent is given. 
Regard the given function as a sine or tangent, find the corre- 
sponding angle as above, and subtract the result from 90°. 

Thus, to find cot -.4750. The next less tangent in the table is 
.4734, which = tan 25^ 20', and the number from column D is 



50 DIRECTIONS FOR USING TABLES. 

3.6. 4750 — 4734 = 16. 16 -^ 3.6 = 4.4. 25° 20' + 4'. 4 
= 25° 24'.4. 90° — 25° 24'.4 = 64° 35'.6 the required angle. 

D. To find the reciprocal of a given number. 

First method. — Beginning with the first significant figure of the 
number, find the first two figures in the column at the left, and the 
third figure at the top. Take the number thus found from the 
table, as also the number in column D on the same horizontal line. 
Multiply the numbers from column D by the fourth figure of the 
given number, with the additional figures, if any (regarded as 
decimals), and subtract the result from the number taken from the 
table. 

Second metliod. — Find in the body of the table the number 
next greater than the given number, and, annexing ciphers to the 
difference, divide it by the divisor D at the right. The reciprocal 
of the given number will then be found as follows : The first two 
figures at the left, the third at the top, and the remaining figures 
in the quotient of the above division. 

Note that in the table of reciprocals a figure 1 is to be supplied 
before the first figure of each number in the columns headed 1, 2, 3^ 
etc., from the line beginning 50 to and including the line beginning 
99. In the two succeeding lines, ivhich conclude the table, the figure 
9 is to be supplied in the same way. 

E. The use of the table of reciprocals in the solution of 
problems. 

When the process of division is to be performed, especiall}^ if 
the divisor or denominator consist of several factors, it is usually 
most convenient to multiply instead by the reciprocals of these 
factors found by Table VII. This method may be employed 
when a sine or cosine occurs in the denominator, but when a 
tangent or cotangent is a divisor the corresponding function of 
the complementary angle can be found from Table VI. and used 
as a multiplier, since the tangent and cotangent are reciprocals. 

When, however, a tangent or cotangent is very large it will 
be more correctly obtained by finding from Table VII. the recip- 
rocal of the complementary function than by interpolating in Table 
VI. Thus, the true value of tan 88° 37'.6 is 41.71; Table VI. 
gives 41.81 ; but tan 1° 22'.4 is .0240, the reciprocal of which, by 
Table VII., is 41.67. 



DIRECTIONS FOR USING TABLES. 51 

To place the decimal point in a reciprocal found by either of the 
above methods. 

If the integral part of the given number, whose reciprocal is to 
be found, consists of only one figure, then its reciprocal is entirely 
fractional, and the decimal point is to be put before its first figure. 
But if more than one figure precedes the decimal point in the given 
number, prefix to its reciprocal, as found from the table, as many 
zeros, less one, as there are figures in the integral part of the given 
number, and put the decimal point before the whole. Or, if the 
given number is a decimal fraction, with no integral part, point 
from the left of its reciprocal a number of figures one more than 
the number of zeros which follow the decimal point in the given 
number before the first significant figure. 

When the first figure of a given number is 5, 6, 7, 8, or 9, the 
first method of finding the reciprocal will ordinarily give the most 
accurate result. When the first figure is 1, the second method is 
best. If the number begins with 2, 3, or 4, it may sometimes be 
expedient, for the sake of accuracy, to multiply the given number 
by 2 or by ^ before finding the reciprocal, and then multiply the 
reciprocal found by the same factor. 

The number found in column D represents the difference between 
successive tabular numbers in the same horizontal line with itself, 
and is strictly accurate only for the middle column. In interpo- 
lating between numbers in other columns, in those parts of the 
table where the value of D changes rapidly, the difference between 
two successive numbers in the table may be found by subtraction 
and used instead of D. The same remark applies to the table of 
tangents, except that in that table the difference found by sub- 
traction must be divided by 10 for the value of D. This method 
must be used for the tangent of all angles greater than 72°. 

Examples. 

1. Find the various functions of the angle 40° 7' 30". 

Ans. Sin = .6445; tan = .8428; cos = .7647; cot 
= 1.187; cosec = 1.5516; sec = 1.3077. 

2. Find the functions of 137° 24'. 

Ans. Sin = + .6769 ; tan == — .9195, etc. 



52 MEAN ORDINATES. 

3. Find the functions of the following angles: — ll' 35'.6 ; 29° 
58'. 1; 57° 0' 6"; 193° 57'; 314° 30'. 

4. Of what angles is .7000 the sine? 

Ans. Of 44° 25'.7 ; of 135° 34'.3 : of 404° 25'.7, etc. 
Of what the cosine ? 

Ans. Of 45° 34'. 3 ; of 314° 25'. 7, etc. 
Of what the tan ? cot ? 

5. Proceed, as in Ex. 4, with the numbers .5125, — .8170, 
2.000. 

6. Divide 3 by 8.3333. 

Ans. 3 X F-uVin? = 3 X .12000 = 0.36. 

MEAN ORDINATES. 

112. A plane figure is a finite portion of a plane. 

A plane is produced by the motion of a generating line of 
unlimited length upon a directrix also unlimited (§12). A 
finite portion of the generatrix, in its motion on a finite part of the 
directrix, describes a plane figure. 

113. In the motion of the generatrix its direction remains 
invariable, but the length of the part, or ordinate, by which the 
figure is described may be either constant or changing. Thus, in 
Fig. 27, A is an example of a figure described with a constant 
ordinate, while B and C were described by variable ordinates. If 
a figure has been described by a variable ordinate it is important 
to know the mean length of the ordinate. For this purpose the 
portion of the directrix over which the ordinate has moved may 
be conceived to be divided into a number of equal parts, and an 
ordinate extended from each point of division. Then the sum of 
the lengths of these ordinates must be divided by the number of 
ordinates. The quotient is the mean length of the supposed 
number of ordinates ; and if it is possible to reduce this quotient 
to a form equally adapted to an unlimited number of ordinates, 
then the true mean length of the ordinate has been found, for this 
is the mean of all the ordinates drawn from all the points of thnt 



MEAN ORDINATES. 53 

portion of the directrix over which the generatrix has passed. An 
example of this process will be given in § 115. 

114. When such a reduction is impossible, the mean length of 
the ordinate can only be found approximately. See Fig. 28, 
where the ordinates represent the temperatures at New Haven, 
Connecticut, at each of the twenty-four hours of the day. The 
twenty-fourth part of their sum is accepted as the mean tem- 
perature of the day, though it is obvious that the true mean 
temperature of the day is tiie mean of the temperatures, not of 
any twenty-four instants, but of every instant through the day. 

115. The mean length of the ordinate under any portion 
of a right line is the ordinate of its middle point, or the 
mean of the ordinates of its tv70 extremities. 

In Fig. 29, let BD be the given portion of a right line, let AX 
be the directrix, or axis of abscissas, and let EB, FC, GD, be the 
ordinates, drawn to the extremities and the middle point of BD. 
Let the distance EF be divided into any desired number of equal 
parts, and the distance FG into the same number, and from each 
point of division let ordinates extend, terminating in MN. Now, 
if MN have the same direction as the axis, these ordinates will all 
be of equal length (§ 35), and any one of them, or the mean of 
any two, will be equal to the mean of the whole number, however 
great that number be ; hence, in this case, the proposition is evi- 
dently true. But if the direction of MN be different from that of 
AX, then, through the extremity, B, let a line pass parallel to AX. 
Represent the distance EG by cl and the number of ordinates by 
n, then the number of equal parts into which d is divided will be 

n — 1, and the length of each part d. By § 37 that por- 
tion of each ordinate which is above the line BP is in a constant 
ratio to the distance of its lower extremity from the point B. Let 
this ratio be denoted by r. Then, if the ordinate of the point B 
be called y, that of the other extremity, D, must be y -j- dtr, 
and the ordinate of C must be y ~\- ^ dr. Also the length of 

the first ordinate, after that of B, is y -| dr ; that of the 

2 3 
next, y -I dr ; the next is, y -I dr ; and so on. 



54 AKEA. 

These ordinates, beginning with y, constitute an arithmetical 
progression, of which the first term is y, the common difference 

— dr, the number of terms o, and the last term y + clr. 

n — 1 "^ ' 

By the rule of arithmetical progression, the sum of the series is 

"H (2 y + ^^)^ ^^^^ dividing this sum by n, the mean length of 

the ordinates is found to be 2- (2 y + dr) or y + i- dr. By 
inspection of this result we see, jirst^ that it is independent of n, 
and hence applicable to any number of ordinates without limit ; 
hence the true mean ordinate has been found ; second^ it is seen 
that this mean ordinate is identical with the ordinate of the middle 
point, C, which had been previously found to be y + 2 ^^ 5 ^^^^ 
third, the mean ordinate is half the sum of the ordinates of the 

extremities, f or y -f- i" dr = J (y + y + «tr). 



AREA. 

116. The area of a fisrure is the amount of surface it con- 
tains. 

117. The area of any plane figure described with a vari- 
able ordinate is equal to that of the figure which would 
have been described by the same generatrix, moving to the 
same distance on the directrix, but with a constant ordinate 
equal to the mean ordinate. 

Thus, in Fig. 28, if 3M be the mean ordinate of the curve 
BCD, then, in moving to the distance AG on the directrix, it will 
pass over as much surface as does the variable ordinate of the 
curve, for its excess of length in any one portion of its path is 
exactly balanced by its deficiency in another. 

118. Let two figures be described by the same generatrix 
and directrix ; then, if the distances over which the genera- 
trix has moved be equal in each, their areas w^ill be as their 
mean ordinates ; but if their mean ordinates are equal, then 
their areas are as the distances on the directrix over which 
these ordinates have moved: in general, their areas are as 



AREA. 55 

tlie product of their mean ordinates into the distances on 
:he directrix over which they have respectively moved. 

In Fig. 30 let the length of the ordinate AE or BF be unity, 
while that of BG is n ; it is plain that when the generatrix has 
moved over a distance, AB or BC, equal to unit}^ then the area 
Massed over by AE (the figure I) will he to that passed over by 
BGr (the figure K) as M to 1. Also, that when the generatrix has 
moved over a distance CD or m, the area of the figure R will 
be to that of K as m to 1 ; hence the area of R is to that of I as 
mn to 1. Moreover, if any other ordinate q of the same genera- 
ti'ix nioves over the distance p of the directrix, the area generated 
will for the same reason be to I as pc| to 1, hence it will be to R 
as pq to mn. 

119. Let two figures be described by dififerent genera- 
trices and directrices ; then, if each be described by a iinit- 
ordinate moving to a unit's distance on the directrix, their 
areas will be as the sines of the angles which their respective 
generatrices make Avith the directrices. 

In Fig. 31 let G and H be two such figures, their directrices 
liaving been placed so as to coincide. Through any point, B, of 
this directrix let a line pass parallel to the generatrix, AY, and 
through any other point, C, a line parallel to the other generatrix, 
AY', and intersecting the former line in D. Let the distance BC 
on the directrix be denoted by c, and the distances CD and BD by 
a and l> respectively. Also let the angle made with AX by AY 
be called a, and that made by AT' be called 3. Now, the figure 
E, included by the lines BD, CD, and AX, may be considered as 
described by the same generatrix as G-, the mean ordinate being 

— (§ 115), hence the proportion 

area E : area G : : — - : 1. 

Li 

But the figure E ma}^ also be considered as described by the 
same generatrix as H, the mean ordinate being — ; 

hence area E : area H : : *—~ : 1. 



56 AREA. 

Since the extremes in the two proportions are ahke, the means are 
inversely proportional, whence 

^ _, ae be 

area G : area H ::—-:-— • 

But -^ : -^ : : a : b, 

and a : b : : sin a : sin i3 (§ 53). 

Therefore area G : area H : : sin a : sin |3, 

which was to be proved. 

120. The unit of area is the amount of surface described 
by a unit-ordiiiate, passing over a unit's distance on the 
directrix, when the sine of the angle of tlie generatrix and 
directrix is also unity; i.e., when those lines are perpen- 
dicular to each other. 

121. The numerical expression for the area of any plane 
figure is the continued product of the mean ordinate with 
which it is described, the distance on the directrix over 
which the ordinate moves in describing it, and the sine of 
the angle which the moving line makes with the directrix. 

For in Fig. 30 it has been shown that the figure R is to I as 
mil is to 1, but (§ 119) the area of I is to the unit of area as 
sin fo is to 1, hence 

area R : 1 : : mil sin u) : 1, 
or area = mn sin <t». 

122. The area under that part of a right line included between 
any two points x', y', and x", y", is 

^(x"-x')(y" + y')sinc., 

where w represents the angle of the axes. 

It is here assumed that the axis of abscissas is the directrix, 
and that the axis of ordinates has the direction of the generatrix. 
The theorem is but an application of that of § 121, for the mean 
ordinate is ^ (y" + y') (§ 115), and the distance to which it 
moves on the axis of abscissas is plainl}' x " — x '. 

Examples. 
In each of the following examples the coordinntes of two points 



AREA. 



57 



are given, to find the area under the rectilinear distance between 
them. In each case the student should not only compute the area 
but construct the figure. 

I. —When w z= 90°. 



Coordinates. Ex. 1st. 2d. 3d. 4th. 5th. 6th. 7th 8th. 9th. 






2 —3 

7 3 

6 2 

8 



5 1 

4 —2 

7 

4 5 



— 2 

— 8 
5 

— 1 



— 9 5 
3 11 

— 2 5 

— 3 3 



— 7 

— 11 

— 1 

— 3 



II.— When oj — G0°. (§ 108.) 



w = 150°. 





Coord hi ates. 


Ex. 10th. 


11th. 


12th. 


13Lh. 




X' = 


3 





2 









5 

7 


8 
2 


4 
5 


8V3 
6 




y"= 


5 


4 


—2 






In example 5th and some of those which follow, one of the ordinates is nega- 
tive while the other is positive. In this case the area computed will be the 
algebraic sum of two parts, a positive and a negative area, one lying above 
and the other below the axis. As these two parts are of opposite signs, their 
algebraic sum is of course their arithmetical difference. If the negative ordi- 
nate be the larger, or if both ordinates be negative, then the computed area 
will be negative unless the abscissas are taken in such an order as to make 
X " — X ' negative. 

123. Two plane figures are said to be similar to one 
another when they differ only in the scale of magnitude 
upon which each is constructed ; or, in other words, when 
the distance between any two points in one figure is in a 
constant ratio to the distance between corresponding points 
in the other figure. 

For example, two maps or plans of the same piece of ground, drawn to dif- 
ferent scales, are similar figures. 

If r denote the ratio of homologous distances (that is, of the 
distances between corresponding points) in two similar figures, 



58 POLYGONS. 

then, since tlie ordinates by which the two figures are described 
must move under like laws of inclination to the directrix and varia- 
tion in length, the mean ordinate ii of the first figure will corre- 
spond to a mean ordinate rn of the second ; while if the distances 
over wliich the ordinate of the first figure moves be m, the cor- 
responding distance in the second figure will be rm. Therefore, 
the area of the former figure will be to that of the latter in the 
ratio of nisi to i'3ii X I'll, or of 1 to r^ Hence, 

The areas of similar figures are to each other in the 
duplicate ratio of their homologous distances. 

POLYGONS. 

124. The boundary-line which separates any plane figure from 
the adjacent surface is called its periphery. The periphery may 
consist of a continuous curve, or it may be in part curved and in 
part straight, like C in Fig. 27, or it may consist of a broken line 
having each of its parts straight. In the last case the figure is 
called a polygon. The periphery of a polygon is also called its 
perimeter. The straight portions of the broken line which con- 
stitutes the perimeter are severally called sides of the polygon, 
and the intersections in which the sides terminate are called 
vertices. 

125. A polygon of three sides is called a triangle; of four sides, 
^quadrilateral; of five sides, Si pentagon; of six, a. hexagon, etc. 

A quadrilateral having no two sides parallel is called a tra.- 
pezium ; if two sides are parallel, a trapezoid; if the four sides 
are parallel, two and two, a parallelogram. In a parallelogram 
the opposite sides are equal to each other (§ 35). li^ adjacent 
sides are equal to each other, the parallelogram is a rhomb; if 
perpendicular to each other, it is a rectcmgle; if they are neither 
equal nor perpendicular, a rhomboid; if both equal and perpen- 
dicular, a square. 

126. In a triangle, or in any kind of a parallelogram, any one 
of the sides may be taken as the base. In a trapezoid, either of 
the parallel sides may be taken as the base. Then, any vertex 
except the two extremities of the base may be taken as the prin- 



POLYGONS. 59 

cipal vertex of the figure. The angle which a side extending from 
the principal vertex to the base makes with the base is called the 
base angle. The angle between the two sides which meet in the 
principal vertex is called the vertical angle. The distance from 
the principal vertex to the base, in a direction perpendicular to 
the base, is called the altitude. The altitude is equal to the pro- 
duct of the sine of the base angle into the side extending from 
the principal vertex to the base. 

127. The areas of triangles, trapezoids, and parallelograms may 
be found by a simple application of the rule for the area under a 
right line. The base is to be taken for the axis of ordinates, and 
the adjacent side extending to the principal vertex, for the axis of 
abscissas. Then, in the parallelogram, the mean ordinate is the 
base, in the triangle half the base, and in the trapezoid half the 
sum of the parallel sides. [The student will give the reason in 
each case, and will also deduce the following propositions : ] 

The area of the trapezoid is the product of the altitude 
into half the sum of the parallel sides, or into the distance 
between the middle points of the sides which are not 
parallel. 

The area of the triangle is half the product of the alti- 
tude and base, or half the product of two adjacent sides 
into the sine of the angle which they make with each other. 

The area of the parallelogram is the product of the alti- 
tude and base, or the product of any two adjacent sides 
into the sine of their angle. 

The area of the rectangle is the product of any two adja- 
cent sides. 

The area of the square is the second power of any one of 
its sides. 

[The last proposition indicates the origin of the use of the word " square " as 
a synonym for "second power."] 

128. To find the area of a trapezium, or of any polygon of more 
than four sides, it is sometimes most convenient to divide the 
polygon into parts by diagonal lines, and compute the area of the 
parts separately. But when the coordinates of the vertices are 
given, or can be found, the following rule may be used : 



60 



POLYGONS. 



Find the area under each side separately ; the algebraic 
sum of these areas will be the area of the polygon. 

Observe that as the formula i (x" — x') (y" + j') sin oj is applied 
to the successive sides, the coor linates of each vertex appear in two of the 
separate computations and no more, being substituted in one of these compu- 
tations for X", j", and in the other for x', y'. 

Thus, in Fig. 32, beghmiiig at the vertex C, and proceeding 
toward the right, the area under the first side is positive ; but as 
the abscissa of the vertex E is less than that of D, the area under 
this side, DE, is negative, because the factor x" — x' is so ; and 
so with all the succeeding areas until the vertex G is reached. 
Now, if the sum of all the negative areas be (algebraically) added 
to that of the positive areas, the entire sum (or arithmetical differ- 
ence of the partial sums) will be the area of the polygon, which, 
in this case, is negative. 

By drawing figures to illustrate all of the following examples, 
the student will convince himself that negative coordinates of some 
of the vertices will not affect the truth of the result. 

Coordinates of Vertices. 



Example. 


1st. 


2d. 


3d. 


4th. 


5th. 


6th. 




X J 


X y 


^ J 


X J 


X y 


X y 


1st . . 


19, 


23, 3 


11, 8 


7, 5 






2d 




9, —5 


13, —2 


1, 3 


— 3, 






3d 




0, 


—6, —8 


—11, —3 


—12, —10 


-4,-12 


+ 3, -9 


4th 




—2, 


6, 


6, 








5th 




13, 1 


9, 7 


27, 12 








6th 




9, 7 


6, 6 


1, 19 








7th 




13, 1 


9, 7 


6, 6 


1, 19 


9 7 


27, 12 


8th 




13, 1 


1, 19 


6, 6 


27, 12 







129. The area of the triangle whose vertices are x', y', x", y", 
and x'", y'", is 
i sin a. [(X" - X- ) (y " -(- y' ) + (X'" - x" ) (y"' + y" ) 

+ (x'-x'")(y' + y'")]; 

or, expanding and rearranging terms, 

area == ^ sin o> [y' ( x" — x'" ) + y" (x'" — x') + y'" 
(x' — x")]. 



POLYGONS. 61 

130. The process of § 128 is much used in surveying. This 
application will be understood from an example (Ex. 1st, below). 
In surveying the field (Fig. 32) the surveyor begins at the vertex 
€, and goes entirely around the field, measuring the length of each 
side, and the angle which it makes either with a meridian or with 
some line that passes through the vertex C and is assumed as an 
axis of abscissas. These measurements he tabulates as in the first 
two columns of the table below, on page 62. The next two columns 
contain the cosine and sine respectively of the angles in the second 
column. The next columns, headed x" — x' and y" — y', are 
computed by multiplying the distances from the first column into 
the corresponding cosines and sines. (When the axis, AX, is a 
meridian, — north and south line, — this difference of like coordi- 
nates of the two extremities of a side is called the northing or 
southing and the easting or westing of the side.) The algebraic 
sums of these two columns separately should each be zero, as the 
student may readily show, and, when such a result is obtained, it 
is a proof of the correctness of the measurement and of the com- 
putation thus far. In the present instance, in consequence of 
some slight error of measurement, the sum of the column x" — x' 
is .0001 and that of y" — y' is — .0003. When such errors are 
not sufficiently large to demand a remeasurement of the field, the 
error is usually distributed to the different sides in proportion to 
their length. The succeeding column is the result of correcting 
in this way the column y" — y'. In the next column, headed y", 
the first number is transferred from the preceding column, while 
each succeeding number is the sum of the number just above it 
and that at the left of it. In the next column, headed y"4~ y'l 
the first number is transferred as before, while each of those which 
follow is the sum of two numbers in the column headed y", 
namely, the one in the same horizontal line, and that in the line 
above. Two more columns are now formed by multiplying the 
numbers from the column x" — x' into the corresponding num- 
bers in the column y" -|- y', and placing the positive and negative 
products in separate columns. Finally, each of these last two 
columns is added, the less sum subtracted (algebraically added) 
from the greater, and the difference divided by 2. The result, 
neglecting its sign, is the area of the field. 





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FLANE TRIGONOMETRr. 63 

Example 2. — The measured lengths of the sides of a field are 
6.26 ch., 7.10 ch., 4.44 ch., 2.58 ch., 5.40 ch. ; and the angles 
which they severally make with the meridian are 56° 30', 145° 45', 
237° 0', 276° 30', and 325°. Required the area of the field. 

PLANE TRIGONOMETRY. 

131. Plane Trigonometry has for its object the solution 
of plane triangles, by which is meant, computing from cer- 
tain known j9a?'fe, (sides or angles,) the unknown parts of 
a triangle and its area. From any three of the six parts of 
a triangle, except the three angles, the other parts and the 
area may be found. 

132. In the usage of trigonometry, the interior angle which any 
two sides make with each other is said to be adjacent to either of 
these sides, included by both of them, and opposite to the third. 
In the following §§, the sides of a triangle will be denoted by a, 
1>, and c, the angles opposite each respectively by a, /3, and ^, and 
the area by A. 

133. All formulae for the solution of triangles are based on the 
following, with which the student is already familiar: 

I. a + ^ + ;'=180°. (§29.) 

II. a : b :: sin a : sin i3 ; 
a : c : : sin a : sin y ; 

b : c : : sin |3 : sin y. (§ 53.) 

III.. A = ^ be sin cc = ^ ac sin |3 = ^ ab sin ^. (§ 127. ) 

134. In special cases, however, the foregoing formulae need to 
be supplemented by certain others which belong merely to the 
theory of angles, and might have been presented earlier but that 
they would have been too widely separated from their application. 
They are deduced as follows : 

Let (i and y be any two angles, then 
)3 = i (^ + r) + i (^ - r) and ;^ ^ X (3 + ^) _ a. (/3 _^). 
Hence (§ 101) 

sin /3 = sin ^{(i + y) cos i. (j3 — ;-) + cos ^ {H + y) sin ^ (/3— r), 
and 



"^ PLANE TRIGONOMETRY. 

Hence ^ 

sin [3 -{-sin r = 2 sin ^ (f3 + y) cos ^((3 — ^) (1) 

and sin/? — sinr = 2cos^ (/3 + ;') sin ^ (13 — y) (2) 

also (§ 102) 

sin (f3-^ y) = 2 sin i(f3 + r) cos K/? + r). (3) 

From (1) and (2) 

sin/3 + sinr : sin/? — sin ^ :: 2 sin ^ (/? + ;-) cos J- (|3 — ;-) : 

2 cos i (f3 + r) sin i(f} — r)', 
whence, dividing the terms of the second ratio by 2 cos ^ (/3 + ;') 

cos J (/3 — ^), and remembering that -?HL^ = tan «, 

cos a ' 

sin f3-\-sin r I sin /3 — sin ;- : : tan J (i3 +;-) : tan J 
(/3-r). , (4) 

More simply, from (1) and (3), 

sin /3 4-sin r : sin (13 -{-y) 11 cos ^ (/3 — ^) : cos i (3 + ^) (5) 
and from (2) and (3), 
sin i3 — sin r : sin {(3-irr) '' sin ^ (l^ — r) t sin ^ (|3 + r) (6) 

135. When any two angles of a triangle are given, the third 
may be found by Formula I., § 133. The process is so simple 
that all the angles may be considered as given if any two are, 
hence the possible cases which may arise in the solution of tri- 
angles are reduced to the four following : 

I. Given the angles and a side ; as, given a, /?, ;', and a, to 
find fo, c, and A. 

II. Given two sides and an angle opposite one ; as, given a with 
a. and l>, to find /3, ^, c, and A. 

III. Given two sides and the included angle ; as, given a with 
h and c, to find i3, y^ a, and A. 

IV. Given the three sides, a, t>, and c, to find a, (3, y^ and A. 

136. Case I. — Given «, /3, ^, and a. 

It is simplest to find one of the unknown sides before finding 
the area. The formulae are, — (see § 133, II. and III.) 

sin a : sin j3 : : a : b 



PLANE TRIGONOMETRY. 65 

sin a : sin ^^ : : a : c 

^ ab sin ;' or ^ ac sin (3 = A. 

If it be desired to find the area directly from the given parts, 
the following formula may easily be deduced from the foregoing; 

a^ sin i3 sin ^ 

sin a 

137. Case II. — Given «, a and b. 

This case is reduced to Case I. by finding the unknown angles. 
The formulae are : 

a : b : : sin « : sin |3 

180° — (a + ^)=r. 
In finding the angle 3 from the table when its sine has been 
■computed, the student will remember that the same sine belongs 
to several angles, for which see Table 11. § 94. But no negative 
value of j3 must be taken, nor any positive value which, substitued 
in the formula 180° — (a-|-j3)=^, will render^ negative. No 
more than two values can possibly satisfy this test, but by apply- 
ing it in each particular example the student will find whether the 
conditions of that example may be met by two different triangles, 
or by only one. When there are two, the values of (3 in the two 
triangles are supplements of each other. 

138. Case III. — Given a, b, and c. 

The area may be found directly by the formula (§ 133, III.) 
^ be sin a = A. 
To find the unknown angles the proportion (§ 133, 11.^ 

b : c :: sin i3 : sin ;' 
is to be taken by composition and division : 

b -[- c : b — C : : sin /3 -|- sin ^^ : sin Z? — sin ;'. 
But, (§ 134, Equation 4) 

«in /3 -|- sin ;- : sin /3 — sin ^ : I tan ^ (jS + ^) : tan ^ ((3 — y). 
Hence b + C : b — c :: tan ^(i3 + ;') : tan ^ (/3 — y). 

In this proportion, the first and second terms are very easily 
found, ns is also the third, since (3 -{- y = 180° — a. Hence the 



6Q PLANE TKIGONOMETRY. 

fourth term is determined, and (3 and y may be obtained from 
the formulae : 

The angles being found, this case is reduced to case I. 
139. Case IV. — Given a, h, and c. 

Since a=180° — ((3 + r), sin a = sin (i3 + r) (§89), 
whence (§ 133, II.) 

sin (/3 + ^) : siuiS :: a : b (i). 

Taking the proportion sin j3 : sin ^^ : : I> : c by composition 

and then by division, we have 

sinjS + sin;- : sin,8 :: fo + c : b (2> 

sin /3 — sin;' : sin i3 :: b — C : b (3) 

From (1) and (2) 

sini3 + si"r • sin(/3-{-r) •• b + c : a (4) 

From (1) and (3) 

sin /3 — sin ^ : sin (i3 -f- r) '' ^ — C : a (5) 

Combining proportions (4) and (5) of the present § with (5) 
and (6) of § 134 by equality of ratios, 

cos^(i3 — ;-) : cosJ-(i3 + ^) :: b + c : a (6) 

an' sin^(i8— r) : sin ^ (13 -\-y) :: b — c : a (7> 

But ^ (/3 + ^)=:i (180° — a) =90° — ^ a, 

hence cos |- (3 -|-^) =&in ^ «, 

and sin ^ (i3 + ;') = cos J- a. 

Substituting these values in the proportions (6) and (7), and con- 
verting them to equations, they become : 

cos^(/3-r) = '^i^sin^a (8) 

a 

I) f» 

sin ^ {13 — r) = cos ^ a (9) 

a 

Squaring these values of cos ^ (13 — ;') and sin ^ ((3 — ;') apd 
adding the results, we have, because the sum of the squares of the 
sine and cosine of any angle is unity (§ 48) 

1 = (^ + ^>' .in' i a + ^^-^' COS^ i a. (10) 

a '^ ' a^ 



PLANE TRIGONOMETRY. 67 

This equation may be put in a form containing only one func- 
tion of ^ a, either by substituting 

1 — cos*^ ^ a for sin^ \ a or 1 — sin^ ^ a for cos' ^ a. 
The latter substitution gives 

^ (b + c)'' . . , (b — cy (b — cy . , , 

4bc . . ^ 1 (b — c> 
whence t" sm^ i a z= 1 — ^ ^-^^ 

or sin J. arr: ^b^ ^^^^ 

The numerator of this fraction may be factored, and written 
(a — b + c) (a + b — c) ; and if s be put for a + b + c the 

fraction becomes ^ ^-^ -^ or, dividing both nume- 
rator and denominator by 4 

sin« ^ a= ^ ^^ i_lA \ 

be 

whence extracting the square root 






(12) 



By pursuing a similar line of reduction from the point where 
1 — cos^ ^ a is put for sin* ^ a in equation (10), the student will! 
obtain |^~7^ T" 

2(2-^) . 
'^'^" = J b^— (13) 

Dividing (11) by (12) (see § 49), 
tan -J a = 



(I-**) (|-«). 



s /s 



(l-») 



(14> 



If equation (11) be multiplied through by 2, and then each 
member subtracted from unity the first member, 1 — 2 sin' \ a be- 
comes equal to cos a, since cos 2 a = 1 — 2 sin' a (§ 102). 

Hence cos a=: 1 — ^'~ ^V~^>' 

2 be 

b* + e' — a' 
or ^^""== 2 be (1^) 



68 PLANE TRIGONOMETRY. 

Multiplying (12) and (13) together, and the product by 2 
remembering that 2 sin ^ a cos ^ a = sin a (§ 102), we have 



^^— ifcjid-^xi-^xi-c). (16) 

And substituting this value of sin a in the formula A = 
J- be sin a ; 



^=jl (!-*») (!-»») (l-«)- (17) 

Equation (17) is the formula for finding the area directly from 
the given sides, The angle a may be found by using either of the 
equations (12), (13), (14), (15), or (16) ; but of these (14) will 
usually be found the most convenient. When one angle has been 
found, the others may be computed by the methods of preceding 
cases, or, better, the process of finding the first ano-le may be 
repeated, using the analogous formulae. 



, ,, l(f-«)(|-«) 

tan ^ j3 = |_r 



and tan ^ r = 



(|-«) (!-»») 



(|-«) 



140. In the first three of the foregoing cases, the solution is 
much simplified when the value of a known angle is 90°. The 
formulae for solution, which are here appended for the sake of 
completeness, may in that case be derived directly from the defi- 
nitions of the functions of angles. The student should give the 
proof in detail, and may also reduce the formulae of the preceding 
§§, by substituting 90° for one of the known angles. In the fol- 
lowing formulae it is assumed that the angle a is a right angle. 

Case I. — Given [3 and a. 
y=^d0° — !3 b = a sin /?. c = a cos (3, 
Case II. — Given 13 and b. 



PLANE TRIGONOMETRY. 



7- = 90° 


-^P. a- .*• . 
sm (3 


Case III. 


— Given (3 and c. 


^-=90° 


_/3. a==— ^ 
cos 13 


Case IV. 


— Given a and b. 


sinjS = 


b b 

- . cos ;' = — 
a ' a 


Case V. - 


— Given b and c. 


tani3 = 


b , c 




«= ^ . 



c = b cot (3. 



c tan [3, 



c = V(a + b)(a-b). 



a = V b' + c% or 



sin 13 

EXAIMPLES. 

141. (A figure should be drawn for each.) 

1. Given a = 1760, P = 85° 15', y = 83° 45'. 

2. Given a = 5, b = 5, « = 30°. (See Fig. 33.) 

3. Given [3 = 84° 47' 38", y = 40° 10', a = 145. 

4. Given b = 5780, c = 7639, (3 — 43° 8'. 

5. Given a = 37, b = 13, c=40. 

6. Given a = 445, b = 83, ^ = 87° 55'. 

7. Given a = 73, b = 55, a = 90°. 

8. Given a = 212.5, b = 836.4, a r= 14° 24' 25". ' 

9. Given c = 520, y = 66° 2' 52", a = 569. 

10. Given a r= 90°, a == 401, b = 399. 

11. Given a =r 353, b = 272, c = 225. 

12. Given a = 101° 17', ft =z 78° 43', c = 125. 

What absurdity is involved in the statement of this problem? 

13. Given a =i 67° 22' 49", (3 =z 45° 14' 22", b = 100. 
Show from the formulae used in this example, that if two angles. 

of a triangle are equal, the sides opposite them are also equal. 

14. Given a = 79, b=:31.6, jS = 98° 53'. 

This example illustrates the proposition that the greatest angle 
of any triangle is always opposite the greatest side, which may be 



70 PLANE TRIGONOMETRY. 

proved as follows : — Let (3 be the greatest angle, and y either of the 
others ; then sin [3 > sin ^, whether (3 be greater or less than 90"*, 
for as angles increase from 0° to 90°, the sine increases uninter- 
ruptedly ; hence if i3 <^ 90°, then (3 and y are two angles each less 
than 90° of which (3 is the greater ; hence sin (3 > sin y. But if 
(3 > 90°, then 180° — i3 < 90° ; and since 180° — [3 z= a + y, 
180° — i3 > r ; therefore, sin (180° — ^) > sin y But, sin (180° 
— (3) =. sin (3^ hence again sin i3 ^ sin y. But the sides t> and c 
are in proportion to sin [3 and sin ;', hence b > C. 
15. Given a = 4.6% b = 17, c=17. 

Show that when two sides of a triangle are equal, the opposite 
angles are equal. 

16. Show that when two sides of a triangle are equal, the line 
which divides their included angle into two equal parts is perpen- 
dicular to the third side and divides it equally. 

17. Given a = 73° 12', a = 12, b = 87. 

18. Given a = 13, and all the angles equal to each other. 

19. Given a = 43, b = 68, c = 25. 

Show that the half sum of the sides of a triangle must always be 
greater than any side, and so prove that the sum of any two sides 
is greater than the third, and their difference less than the third. 

20. A tower 150 feet high, standing on a horizontal plane, casts 
a shadow 75 feet long. Find the sun's altitude. 

21. A tower stands by a river. A person on the opposite bank 
finds the elevation of the top to be 60° ; receding 40 yards in a 
direct line from the tower he finds the elevation 50°. What is the 
breadth of the river ? 

22. Wishing to know the distance between two inaccessible 
objects, A and B, I find a level place from which both can be 
seen, and measure then a distance CD equal to 134 feet, C is N. 
5° E. from D ; A is N. 45° 17' E. from D, and B is N. 70= 41' E. 
from D ; B is due east from C, and A is N. 80° E. from C. What 
Is the distance and direction of B from A? (See Fig. 34.) 

23. Th,3 planet Mercury is said to be at its greatest elongation 
when lines drawn from it to the sun and the earth are perpendicu- 
lar to each other. On March 29, 1879, Mercury was at its greatest 
elongation, and, as seen from the earth, was at an angular dis- 
tance of 18° 57' from the sun, the earth at that time being at a 
linear distance of 92,700,000 miles from the sun. How far was 
Mercury from the earth? 



THE CIRCLE. 71 



THE CIRCLE. 



142. A point moving in a plane, so as to retain always a con- 
stant distance from a fixed point, describes a clrde. The fixed 
point is called the centre^ and the constant distance the radius. 
Any portion of the path of the point is called an arc^ and the rec- 
tilinear distance between the extremities of an arc is its chord. 
When two lines pass through the centre of a circle, the angle which 
they make with each other is said to be subtended by the arc 
which they intercept. 

143. The circle is a curve of uniform curvature^ so that any 
two arcs described with equal radii may be so placed as to coin- 
cide throughout the whole extent of the shorter arc. For when 
the centres coincide, and the extremities of the shorter arc lie on 
the longer one, intermediate points of the two arcs cannot fail to 
coincide without being at unequal distances from the centre. 

144. In § 18, the angle between two lines is conceived to pass 
through all successive values from 0° to 180° by the rotation of 
one line upon a point of another. In this rotation, any point of 
the moving line (AC, Fig. 35) describes an arc of a circle, and 
it is obvious that this arc increases in the same ratio with the 
angle. Or, 

Arcs of the same circle are to each other as their sub- 
tended angles. 

145. To square the circle is to find a numerical expression for 
the area of the inclosed figure, and to rectify it is to find the 
length of the circular periphery, the radius in each case bein<y 
known. These problems are of great celebrity in the history of 
mathematics, but it has been demonstrated that the precise values 
of neither area nor length can be found, though an approximation 
may be made to any required degree of exactness. The simplest 
method of approximation is by means of polygons whose vertices 
lie in the curve, hence called inscribed polygons. 

Let the entire curve (Fig. 36) be divided into n equal parts, 
the chords of these n arcs will constitute the perimeter of a poly- 
gon of n sides. Let lines pass through the centre, and extend to 



72 THE CIRCLE. 

each of these points of division. Then any one of these lines will 

360° 
make with the line next to it an angle of It is obvious that 

the greater the number m is, the more nearly will the peripheries 

and areas of the two figures coincide ; although for any finite 

value of M, however large, the polygon must have a smaller area 

than the circle. Now the polygon is made up of ii triangles, in 

each of which two sides and the included angle are known. The 

length of one of these sides being equal to the radius, may be 

360° 
called r, while the included angle is The remaining angles 

may easily be found, for since they are equal to each other (§ 141, 

Ex. 15), and their sum is 180° ? each of them is equal to 

180° 
90' Hence the remaining side may be found by. the pro- 
portion, 

sin 



x 180°\ S60° 
in (90° -) : sin :: r : the third side (§ 136). 



Now, sin (90° — —_^ 



180°\ 180= 

cos ; 

n 



, . 360° „ . 180° 180° ._ ,__. 

and sm = 2 sm cos (§ 102) ; 

n II II ^ -" 

and by substituting these values in the proportion, the value of 

180° 
the third side is easily found to be 2 r sin 

The area of the triangle (Formula III. § 133) is ^ r^ sin 



n 

As there are n triangles, the perimeter and area of the whole 
polygon will be found by multiplying these results by n ; 

whence perimeter = 2 rn sin - — - 

A 1 2 . 360° 

and area = 4- r n sm 

n 

By these formulae may be com.puted the perimeter and area of 

inscribed polygons, having any required number of sides. For 

instance, if ii = 6, perimeter = 2 r X 6 sin 30°, or 2 r X 3, while 

the area = i r' X 6 sin 60° = p' X f V 37 If ii = 12, perimeter 

= 2 r X 12 sin 15°, and area = ^ v' X 12 sin 30° = 3 r\ 



THE CIRCLE. 



73 



By the aid of the table of sines the following may be computed 
and indefinitely extended : 



Name of Polygon. 



Triangle 
Hexagon 
Dodecagon 



Value 
of m. 


Perimeter 
= 2 r X 


Area = r2X 


3 


2.59808 


1.29904 


6 


3.00000 


2.59808 


12 


3.10582 


3.00000 


24 


3.13263 


3.10582 


48 


3.13935 


3.13263 


96 


3.14104 


3.13935 


192 


3.14145 


3.14104 


384 


3.14156 


3.14145 


768 


3.14158 


3.14156 



Here the successive values of n are each double the preceding, 
but the values of the perimeter and area are far from increasing 
at the same rapid ratio. In fact their increase, rapid for the first 
two or three terms, becomes much less so as we proceed, until we 
find that by doubling the number of sides the perimeter is increased 
only in the ratio of 3.14158 : 3.14156 while m is no greater than 
768. This is due to the principle (explained in § 110), that the 
sines of very small angles are nearly in the ratio of the angles 



themselves. 
180 



Hence, if n and m are very large numbers. 



180^ 
n 



and 



in 



being very small angles, we have the proportion, 



But 
Hence 



. 180° 

sin 

n 

180° 



180^ 



sm 



m 



180^ 
n 



180= 



180° 



sm 



n 

180^ 
n 



m 



n. 



sni 



180' 
in 



or 



m sm 



180^ 
m 



= n sin 



; in 

180° 



Thus if m = -^ n, we have ^ n sin 



a 

180° 



in 



or 



nearly. 



nearly; 

nearly. 
. 360° 



n sin 



n 



nearly 



74 THE CIRCLE. 

180° 
equal to n sin , so that the formula for area, when n is very 

180° 
great, may be written r^n sin • Or, without restricting the 

180° 

value of in to 4- n, it may be said that the formula n sin ' 

n 

when n is increased, rapidly approaches a fixed value, and may 

be brought within any required closeness of approximation to this 

value by sufficiently increasing n. This fixed quantity, the value 

180° 
of n sin ' when n = oo, is denoted by the symbol tt, and as 

the polygon is brought into coincidence with the circle by the 
indefinite increase of n, we have 
Periphery of circle 

180° 
( = 2 VVL sin [when u = oo] ) c= 2 tti*, 

Area of circle 

i = ^ r'n sin = r^n sin [when n = oo] ) = 7rr'. 

146. The numerical value of tv may be found to six decimal 

180° 
places by giving n in the formula n sin a value no greater 

n. 

180° 
than 10800. then becomes 1', and (§ 110) 

10800 sin l'r= 10800 X .0002908882 = 3.1415926. 
The value in common use, when great accuracy is not required, 
is 3.1416, and frequently the value ^- is found to be exact enough 
in practice. The computation of tt has, however, been carried to 
hundreds of decimal places. The value to fifty places is 
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 
37510. 

147. The arc which is equal in length to the radius subtends an 

180° 
, angle of -^^' or 57° 17' 44".8. 

TV 

For since an arc whose length is 7rr, or half the periphery, sub- 
tends an angle of 180°, the arc whose length is V subtends an 

angle of (§ 144). 



I 



THE CIRCLE. 75 

148. Another approximate computation of the circular area, independent of 
the foregoing, may be made by the method of §^ 114, 121. Let rectangular axes 
pass through the centre of the curve (Fig. 37) ; they will divide the figure into 
four equal parts. The distance on the axis of abscissas over which the genera- 
trix must move to describe one of these quarters is equal to the radius. Let this 
distance be divided into any number, say twenty, of equal parts or units, then 
the ordinate at any point of division may be found by subtracting the square of 
the abscissa of the point from the square of twenty, and extracting the square 
root of the remainder (§ 47). Finding in this way the length of every second 
ordinate, ten results are obtained : 

Vsyg, Vsel, \/m, VsET, Vm, \/279, \/23r, \/i75, Vm, and V^-, 

or 

20.0, 19.8, 19.4, 18.7, 17.9, 16.7, 15.2, 13.2, 10.5, and 6.2. 

The sum is 157.6, and the mean of these ten ordinates 15.76, or 0.788 of the 
radius. Taking this as the mean ordinate, the area is found by multiplying 
the ordinate by the radius (on the axis of abscissas), and the product by four. 
If the true value of the mean ordinate be computed from the formula A = 7ri*^ 
using the value of tz found in ^ 146, it will be found that it diifers from the 
above mean of ten ordinates by only 0.003 of the radius. 

149. If two circles are described with different radii, their 
peripheries are as the radii, and their areas as the squares 
of the radii. 

For let the radii be r and r', then the peripheries are 2 rr and 
2 7rr',and the areas -r^ and tzv'^. 
But 2 Trr : 2 -r' : : r : r', 

and 7rr^ : nv'^ :: r* : r'^ 

Also, 

Two arcs subtending equal angles are as the radii with 
which they are described, since they are proportional parts 
of their respective circles. 

Examples. 

1. If the equator of the earth is a circle, whose diameter is 
7925.6 miles, what is its circumference? 

2. If the meridian passing through Colorado Springs were a 
circle of the same diameter, what would be the perpendicular dis- 
tance of that place from the axis of the earth, its latitude being 
38° 51' ? 

3. The longitude of Colorado Springs is 23° 18' W. from Washing- 



76 THE CIRCLE. 

ton, while the latitude of the two places is nearly equal : Required^ 
the distance which would be traversed by a person journeying due 
east from Colorado Springs to Washington. 

4. The planet Saturn is surrounded by several flat rings, the 
outermost of which is estimated to have a circumference of 527,- 
000 miles, with a breadth of 10,000 miles: Required, the area of 
this ring; also its distance from the surface of the planet, the 
diameter of the latter being 70,500 miles. 

6. A circular pond has an area of a quarter of an acre ; find the 
area of a driveway fifteen feet wide around the pond. 

150. The ratio of the arc to the radius is a new function of the 
subtended angle, comparable with the sine, cosine, etc. This 
function of any angle a may be denoted by the expression arc a, — 
an expression which will represent the length of the arc only when 
r = 1. The arc differs from all the other functions in this 
important respect, that the arcs of two angles are in the same 
ratio as the angles themselves. (See Fig. 38, where the ratio 

BP CM DN ER ^, ^ 1. . 

- = - = ■ = -^ the numerators being arcs^ not 

AB DM AN AE 

rectilinear distances. This constant ratio of arc to radius is the 
function arc a.) 

151. The ratio of the two functions, sine and arc, of a given angle is some- 
times required. K n denote the ratio of the given angle a to 180°, 



sm a 



then i^S'J he written 



180^ 
n 



180°. 
arc a arc - 

If both numerator and denominator be multiplied by r, 

. 180° 
r sm • 

this fraction becomes — 



180^ 
r arc 



n 

where the denominator is the length of the arc. Now if each term be again 
multplied by n the denominator becomes the length of that arc which sub- 
tends 180°, that is. Trr. Hence the fraction may be written 

. 180° . 180° 

rn sm n sm • 

n_ or n 

> ■ 

7rr "^ 



THE CIRCLE. 77 



In this form the value of the ratio ?1B^ — maybe found for any angle a by refer- 

arca *' ^ a j 

•ence to a table of sines. 

Thus, if a = 90°, ii==2, and as sin 90« = 1, we have, 

180*^ 



n sm 

sin 90° 1 _ 2X1 



.63662. 



arc 90° :: tt 

sin a 



If a 57° 17' 44"-8, n = tt and 



arc a 



sin a 6X4- 3 
If a = 30°, n == 6 and^^ = -^^ - = .95493. 

sin 180° 
When a = 0**^ n = oo , and the numerator, n — — becomes equal to 

Tt C? 145), hence the fraction is f. or 1. That is : the ratio when a = 0, 

^^ -" ,r arc a 

is equal to unity. 

152. In Fig. 39 the relation of the circle to all the functions of 
an angle is shown. The angle a is represented in four different 
values, — one between 0° and 90°, one between 90° and 180°, one 
between 180° and 270°, and one between 270° and 360°. Rect- 
angular axes pass through the centre of the circle, and of these 
the axis of abscissas may, for the present purpose, be called 
simply the axis. The point O, in which the axis intersects the 
curve at the right, is called the origin of axes; and the point Q, at 
the intersection of the axis of ordinates with the upper portion of 
the curve, is the secondary origin. The point B, C, D, or E, is 
called the extremity of the arc, and the line AB, AC, AD, or AE, 
the hounding line. The radius is denoted by the letter R. 

Angle a is the angle made with the axis by the bounding line. 

r arc a == the arc OB, OC, OD, or OE, measured from the 
origin of arcs around to the extremity of the arc in a direction 
contrary to the motion of the hands of a watch. (See § 60.) 

r sin a = FB, GC, HD, or IE, the ordinate of the extremity of 
the arc or the distance from the axis to the extremity of the arc, 
in a direction perpendicular to the axis. When sin a is positive, 
i.e., when a is between 0° and 180°, this distance will be measured 
upward, but when sin a is negative, or a between 180° and 360°, 
it will be measured downward. 



r cos a = KB, LC, MD, or NE, the abscissa of the extremity 



78 THE CIRCLE. 

of the area, or the distance from the axis of ordinates to the 
extremity, in a direction parallel to the axis AX. This distance 
is measured to the right when cos a is positive, i.e., when a i& 
between 0° and 90°, or between 270° and 360°, and to the left 
when a is between 90° and 270°. 

r tan a = OP, OR, OP, or OR, the distance from the origin of 
arcs to the bounding line in a direction perpendicular to the axis. 
This distance is measured upward when tan a is positive, i.e., 
when a is between 0° and 90°, or between 180° and 270°, but 
downward when tan a is negative. 

r cot a = QV, QZ, QV, or QZ, the distance from the secondary 
origin to the bounding line in a direction parallel to the axis, 
measured to the right when the value of a is such as to make cot a 
positive, but to the left when cot a is negative. 

The reciprocals of the cosine and sine of an angle are called the 
secant and cosecant repectively (abbreviated sec and cosec). The 
secant of an angle has always the same sign as the cosine, and 
the cosecant the same as the sine. 

r sec a = AP, AR, AP, or AR, the distance from the centre, 
measured on the bounding line, to the point where that line meets a 
line perpendicular to the axis and passing through the origin of axes. 
This distance is measured toward the extremity of the arc (the 
direction of the positive radius) when sec a is positive, but from 
the extremity when sec a is negative. That either of the above 
distances, AP for example, is equal to r sec a, may be shown a^^ 
follows : — 

AP' = A0'+0P2(§ 47) =^'' + r^t3Ln^a = r^ (l + tan2a) 
^ ,. (1 + g!!^) (§ 49) = r- C' ^ + ""^ ^ > 

^ ' cos' a/ ^ ' \ COS^ a / 

= r' . -^ (§ 48). 

COS a 

^ ^ ^ 1 

Hence AP= r . -— — =r sec a. 

COS a 

In the same way prove the following : — 

1» cosec a = AV, AZ, AV, or AZ, the distance from the centre, 
measured on the bounding line, to the point where the line meets 
a line parallel to the axis and passing through the secondary origin. 



miscellanp:ous examples. 7^ 

This distance is measured toward the extremity of the arc when 
cosec a is positive, but from the extremity when cosec a la 
negative. 

MISCELLANEOUS EXAMPLES. 

1. A ladder, whose foot remains in a given position, just 
reaches a window on one side of a street, and when turned about 
its foot just reaches a window on the other side. If the two posi- 
tions of the ladder are at right angles to each other, and the heights^ 
of the windows 36 and 27 feet respectively, find the width of the 
street and the length of the ladder. 

2. From the top of a hill are observed two consecutive mile- 
stones on a straight horizontal road running from the base of the 
hill. The angles of depression are found to be 45° and 30°, 
Required the height of the hill. 

3. Prove that tan ^3 + cot |3 = -^— -. 

sm p cos ^ 

4. Show that (cos'^e — 1) (cof'l9 + 1) = — 1. 

5. Show that tanV — tan'j8 = ^^^'^ ~" ^^^^ 

cos^jS cosV 

6. What angles have cosines equal to their cotangents? 

7. Show that tan r + tan |3 = ^^^ (?^ + ^) . 

cos X cos ^ 

8. Show that cot (6> + 45°) = ^^^^T! ' 

^ ' ' cot <9 + 1 

« c^^ .. , COS 27° — sin 27° 

9. Show that ^^^^^.^^.^^^, = tan 18°. 

10. Show that cot — 2 cot 2 = tan Q, 

11. If the angle ^ of a triangle be 120° show that c'= a^-f^l^^ 

12. The sides of a triangle are in arithmetical progression, and 
one angle is 90°. Required the other angles. 



80 GEOMETRY OF THREE DIMENSIONS. 

13. The sides of a triangle are in geometrical progression, and 
one angle is 90°. Required the other angles. 

14. Two buoys, B and C, in a harbor, are 500 yards apart, 
while a third, A, is 1,000 yards from B and 700 yards from C, 
A boatman desires to know the position of his boat, from which he 
sees C in a line directly behind A, while the line to B makes with 
the former an angle ol" 18°. 

15. An observer at Colorado Springs notices the sun going 
down just behind the top of Pike's Peak, at a time when its alti- 
tude above the horizon is 8° 47'. Required the elevation of the 
mountain above the level of the town, assuming that the horizontal 
distance is ten miles. 

16. The length AB of a rectangular sheet of paper is a inches, 
while its breadth, AC, is b inches. The paper is folded so that 
vertex A lies on the side CD, the crease passing through the 
vertex B. Required the area of the part folded down. 

GEOMETRY OF THREE DIMENSIONS. 

151. In the foregoing §§, since § 17, the lines, angles, etc., 
which have been considered have been limited to a single plane. 
This restriction is now to be removed, and the remaining §§ will 
treat of the relations of lines situated in different planes. 

In all applications, therefore, of preceding propositions to the 
reasoning of the following §§, the restrictions under which those 
propositions were demonstrated must be remembered, and no 
wider signification must be allowed than that originally intended. 
Thus, the proposition of § 31 means no more than this: *'Two 
lines are parallel if they are perpendicular to the same line, and lie 
in the same plane with it and with each other." 

152. But the theorem of § 26, though introduced after this 
restriction was imposed, does not depend upon it for demonstra- 
tion. Whenever two or more lines which are parallel to each 
other are intersected by other lines also parallel to each other, 
the angles at any one intersection are equal to those at any other, 
even though the plane of the two lines which meet in one intersec- 
tion contains neither of the lines which meet at the other. For 



GEOMETRY OF THREE DIMENSIONS. 81 

the difference between two directions must remain unchanged as 
long as the directions themselves remain so. 

153. If from any point of a plane a line be drawn parallel 
to a given line of the plane, it will lie wholly in the plane. 

For a line may be drawn in the plane, parallel to the given line, 
through the given point ; and since two different lines cannot ex- 
tend from the same point in the same direction, 710 other parallel 
to the given line can be drawn through the given point. 

154. As a line is designated by naming two points upon it, so 
the position of a plane is sufficiently determined by naming two 
lines as contained in it, which either intersect or are parallel. For 
if they intersect, one may be used as the generatrix and the other 
us the directrix of the plane ; and if they are parallel, either may 
be the directrix, while the generatrix is any line joining a point of 
one of them with a point of the other (§§ 14, 15). 

155. The system of Cartesian coordinates with which the student 
has become familiar is rendered applicable to points, etc., not lying 
in one plane, by the use of three converging axes instead of two, 
whence the name " Geometry of Three Dimensions." If AF, AG- 
(Fig. 40), be the axes of coordinates for the plane in which they 
are situated, then AH may be the third axis, provided it does not 
lie in the plane of the other two. 

156. Conceive right lines to extend in the plane of AF and AG-, 
through the point A, in an indefinite number of directions. AH 
cannot coincide with any of these, else it would also lie in the 
plane of AF and AG. Hence it must make a finite angle with 
each of them. In other words, of the angles made by AH with 
the various lines which it meets in the plane there is one angle 
greater than zero, which is the minimum angle, or such that no less 
angle can be made with AH by any line of the plane. This angle, 
whatever be its value, is called the inclination of the line AH to 
the plane of AF and AG, and will be denoted by the letter i. Let 
AM (in the figure) be a line of the plane of AF and AG, which 
makes with AH the angle c, 

157. In the same figure, the line AM' which extends in the direc- 
tion opposite to AM in the plane of AF and AG, must make with 
AH an angle of 180° — c So, if AS be any other line of the plane 

6 



82 GEOMETRY OF THREE DIMENSIONS. 

of AF and AG, which passes through A and makes with AH an 
angle z, the angle of AS' with AH is 180° — x. But as x was not 
less than ;, therefore 180° — y. is not greater than 180 — i ; in other 
words, AM' makes with AH a maximum angle. If a line passing 
through A be conceived to be rotated about this point, while re- 
maining in the plane of AF and AG, it will change its direction 
by insensible variations, hence its angle with AH, changing in the 
same manner, will pass through all possible values between i and 
180°— ^ 

In such a range of values, the value 90° cannot fail to be included, whatever 
the angle t may be ; hence, whenever a line pierces a plane, some line of that 
plane is perpendicular to the first-mentioned line. 

158. Of mU the distances measured on right lines from 
various points of a plane to any one point of a given line 
which pierces that plane, no one is shorter than that which 
is measured upon a line perpendicular to that line of the 
plane with which the given line makes its minimum angle. 

That is, in Fig. 41, if A'M be that Hne of the plane of AF and 
AG with which AH makes its minimum angle f, P any point of 
AH, and RP a perpendicular to AM through P, then no rectilinear 
distance SP between the plane and the point P can be shorter 
than RP. 

Let QP be the perpendicular distance of P from the line AS, 
let X be the less of the two supplementary angles which AH makes 
with AS, and let /a be the angle between SQ and SP. 



Then 


QP z= AP sin X 


and 


QP = SP sin 11, 


whence 


SP:=AP""^ 
sm// 


Also, 


RP — AP sin c. 


Hence 


SP : RP 


sin X 
• • 


: sin r. 



sm p. 
Now, as neither x nor i exceeds 90°, and x is not less than £, 

sin z cannot be less than sin :, and as sin p. cannot exceed 1, — — 

sm p. 

cannot be less than sin x ; hence -: — is not less than sin £, and SP 

sm /x 

is not less than RP. 



GEOMETRY OF THREE DIMENSIONS. 83 

159. When a line pierces a plane, that line of the plane 
which makes with the former a minimum angle, also makes 
a minimum angle with any other line which lies in the plane 
of the two former lines. 

Thus, in Fig. '42, no line of the plane of AF and AG- can make 
a less angle than AM makes with any line (BK or EC) of the 
plane of AH and AM. 

In the case of BK, a line parallel to AH, the proposition is evi- 
dent. For every line of the plane of AF and AG that passes 
through B is parallel to a corresponding line passing through A, 
and hence makes with BK an angle equal to that which its parallel 
makes with AH (§ 152). Hence, as no line makes with AH a 
less angle than AM makes, none will make a less angle with BK. 

But to prove the proposition for EC, a line extending from any 
point E of AM in a direction not parallel to AH, conceive a line 
parallel to AH to pass through C and meet AM in D. Also, let 
a line pass through C perpendicular to AM and meeting it in 
L. Then no line of the plane of AF and AG makes a less angle 
with DC than that which AM makes. Hence no rectilinear dis- 
tance shorter than LC connects C with the plane of AF and AGr 
(§ 158). 

But if any line of that plane, passing through E, made with EC 
a less angle than AM does, then the perpendicular distance from 
that line to C would be shorter than LC. As this has been shown 
to be impossible, it follows that no line of the plane of AF and AG 
can make with EC a less angle than AM makes with it. 

160. If through any line which pierces a plane, and 
through that line of the plane which makes with the former 
a minimum angle, another plane pass, so that the second- 
named line is common to the two planes, then any line 
drawn in either plane perpendicular to this common line, 
is perpendicular to every line which it meets in the other- 
plane. 

In Fig. 43, if AM be, as heretofore, that line of the plane of 
AF and AG which makes with AH the minimum angle, then AO 
drawn in the plane of AH and AM perpendicular to AM, is per- 
pendicular to every line in the plane of AF and AG which passes- 
through A. 



84 GEOMETRY OF THREE DIMENSIONS. 

For no such line can make with it an angle less than its angle 
with AM (§ 159) ; that is, less than 90°. Also (§ 157), no such 
line can make with it an angle greater than 180° — 90°, i. e., than 
90°. Hence all lines of this plane which meet AO in A make with 
it angles neither less nor greater than 90°, or it is perpendicular to 
them all. 

On the other hand, any line as AN of the plane of AF and AG 
drawn perpendicular to AM is perpendicular to every line of the 
plane of AH and AM which meets it in A. For, let a line be drawn 
connecting any point U of this plane with any point V of the line 
AN. Let there be drawn, moreover, a line UW perpendicular to 
AM ; also a line joining U and V, and a line joining V and W. It 
has been already shown, in this §, that UW" will be perpendicular 
to VW. 



Now UV^ 1= VW2 + WU2 but VW' = VA' + AW* 
hence UV^=z VA7 + AW"^ + Wl?. 



Also AU^ = AW^ + WU' hence UV' == VA' + AtP. 

This equation indicates that AU is perpendicular to AV or AN, 
for if we call the angle of these two lines a and the distance UV, a 
and put b for VA and c for AU, we have — 

But (Equation (15) § 139) 

b^ + c^ — a» 

cos a z=. ^— ^- • 

2 toe. 

Hence cos a = 0, and therefore a. =z 90°, or AN is perpendicular 
to AU. 

161. When a line which pierces a plane is perpendicular 
to every line which it meets in the plane, it is said to be 
perpendicular to the plane. 

Thus AO is perpendicular to the plane of AP and AG, and AN is per- 
pendicular to the plane of AH and AM. 

The foot of the perpendicular is the point in which it pierces the 
plane. 

162. If a line is perpendicular to a plane, every line paral- 
lel to the former is also perpendicular to the same plane. 

For every line drawn in the plane through the point in which it 



GEOMETRY OF THREE DIMENSIONS. 85 

is pierced by the second line makes right angles with that line, 
because it is parallel to some line of the same plane which makes 
right angles with the first line. 

163. Through a given point there can pass only one line 
perpendicular to a given plane. 

For if the point be without the plane, and two perpendiculars 
be drawn from it to the plane, the line joining their feet is at right 
angles to both, while the three lines are in one plane (§§ 154, 161), 
which is impossible (§31). But if the given point be upon the 
plane, and two perpendiculars be drawn from it, then from some 
other point of one of them let a line pass parallel to the other ; it 
will be perpendicular to the plane (§ 162), which is impossible as 
before. 

164. The inclination of a line to a plane is the comple- 
ment of the angle which the line makes with a perpendicu- 
lar to the plane. 

In Fig. 41, if a line PR be drawn from any point P of AH per- 
pendicular to AM, it is (§ 160) the perpendicular from that point 
to the plane of AF and AG-. But the three lines AP, PR, and 
AR are in one plane, hence (§ 30) c is the complement of the angle 
between PR and PA. 

165. Hence, two lines perpendicular to the same plane are paral- 
lel. For the angle between them is the complement of 90°, the 
inclination of either to the plane, and is therefore 0°. 

166. If a line be perpendicular to a plane, any line per- 
pendicular to the first line, and passing through its foot, lies 
in the mentioned plane. 

Let AT (Fig. 44) be perpendicular to AG at A, while AO is 
perpendicular to the plane of AF and AG, and if AT do not lie in 
that plane, let AM be that line of the plane which makes with AT 
the minimum angle ; and through T let a line JT pass perpendicu- 
lar to AM, then (§ 160) it will also be perpendicular to the plane 
of AF and AG, hence (§ 165) parallel to AO, and therefore per- 
pendicular to AT because AO is. Hence we have two perpendicu- 
lars, AJ and AT, from the same point A upon the same line, and 
the three lines in one plane, which is impossible. 



86 GEOMETRY OF THREE DIMENSIONS. 

167. If one of two lines remain fixed in position, while 
the other is rotated about the former, maintaining with it a 
constant angle of 90°, the second line describes a plane per- 
pendicular to the first. 

For, as shown in the last §, it can assume no position not lying 
in a plane perpendicular to the first line through the point of inter- 
section ; and in rotating entirely around that line it will evidently 
pass over the whole of such a plane. 

168. The construction of the system of rectangular Cartesian 
coordinates for the geometry of three dimensions can now be ex- 
plained. (See Fig. 45.) From the origin A of rectangular coor- 
dinates in the plane of AX and AY, a third axis AZ is extended 
in a direction perpendicular to that plane. Now, let a plane per- 
pendicular to AX be formed by the rotation of AY (§ 167), then 
AZ will lie in that plane (§ 166). In the same way, let AX 
revolve about AY, a plane perpendicular to AY will be formed, 
which will also contain AZ. Finally, the plane of AX and AY 
may itself be considered as generated by the revolution of either 
of those fines about AZ. It thus appears that there are not only 
three coordinate axes, but three coordinate planes, each axis being 
a line common to two of the planes, and perpendicular to the third. 
The three coordinates of any point are its distances, parallel to 
each of the three axes respectively, from the plane of the other 
two ; thus, in the figure, the coordinates of a point P are QP, RP, 
and SP. If through Q, R, and S, the points in the coordinate 
planes from which these coordinates extend, lines are drawn in 
these planes parallel to the axes (and hence parallel also to QP, 
RP, and SP), then any two of the three lines QP, RP, and SP 
will be in the same plane with two of the lines so drawn. Thus 
QP and RP are in the same plane with LR and LQ (§ 154), which 
lines therefore meet the axis AZ in one point L. 

(§ 35.) 



Hence LR = QP 


and 


LQ=rRP. 


In like manner 






MS=QP 


and 


MQrrSP; 


Also NS = RP 


and 


NR=SP. 


For a similar reason 






AN^MS==LR, and AM 


= NS: 


i=LQ, and AL 



NRr=MQ. 



THE MENSURATION OF VOLUMES. 8? 

Therefore 

AN = QP, and AM = EP, and AL^SP. 

Hence the coordinates of a point, as in plane geometry, may be 
measured at pleasure upon the axes, as well as parallel to them. 

THE MENSURATION OF VOLUMES. 

169. A volume has been defined (§ 17) as generated by the 
motion of a plane figure along a line not lying in its plane. In its 
motion the plane figure is assumed to retain a constant inclination 
to the directrix upon which it moves. Its form and area may 
remain unchanged, or either or both may change continuously in 
accordance with some fixed law. To determine the volume gener- 
ated in the latter case, it is necessary to know the mean area of 
the moving figure. If a number of points be taken at equal inter- 
vals upon the directrix, and the area of the figure at each of these 
points determined, the sum of these areas, divided by the number 
of the points, is the mean of the area for those points, and if an 
expression for such a mean of areas can be put into a form which 
will still be applicable, when the number of points is extended to 
include every point of the directrix over which the generating sur- 
face moves, then the mean area of the surface has been found. It 
is assumed as self-evident, in accordance with the analogy of areas 
described by variable ordinates (§ 117), that the volume generated 
by a variable area, moving to a given distance upon a fixed line, 
is equal to that generated by a constant figure of the same plane, 
moving to the same distance upon the same line, provided that the 
constant area of the latter figure is equal to the mean area of the 
former. 

170. As in the generation of areas by moving ordinates, so in 
the generation of volumes by moving figures, there are three quan- 
tities whose value affects that of the generated volume : 1st, the 
area of the generating figure ; 2d, the distance on the directrix 
over which it moves , and 3d, the sine of the inclination of the 
directrix to the plane of the generatrix. If in the generation of a 
particular volume, any two of these quantities have values equal 
to the values of the same quantities in the case of another volume, 
while the third quantity has different values in the two cases, then 
the volumes generated will be as the values of that third quantity. 
In particular : — 



88 THE MENSURATION OE VOLUMES. 

171. First. — The volumes generated by different figures, part& 
of the same plane surface, the whole of which moves to a given 
distance along a fixed right line, will be as the areas of the figures. 

Thus, in Fig. 46, if the figure M move to the distance AB, or 
m, upon the axis of X, then the volume I described by the whole 
figure M is to the volume II, described by N, as M to N. This 
volume II described by N moving to the distance m (see the 
figure), may be also considered as generated by the figure P 
moving along the line AZ to the distance AC or p. Its value will 
then be to that of III, described by the figure Q in the latter mo- 
tion, as P to Q. In like manner, volume i is to I as V to M : 

ii : II :: W : P. 

172. Second. — The volumes generated by the same figure, mov- 
ing to different distances upon the same right line, are as the dis- 
tances traversed. 

Thus, if the figure N be stopped after passing over the distance 
AD, or n, the resulting volume IV will be to II as n to m. 

Hence I V : I : : Nn : Mm. 

173. TJiird. — The volume generated by a given figure moving^ 
to a given distance varies directly as the sine of the inclination of 
the directrix upon which it moves to the plane of the figure. 

Thus, in Fig. 47, as AZ is perpendicular to the plane of the 
figure P, the angle £, made by the directrix AZ' with AY (which 
is the complement of the angle it makes with AZ), is its inclina- 
tion to the plane of the figure (§ 154). Now, if the distances 
AL, AM, and AP be called 1, m, p, respectivel}^ then the area of 
the figure N is Ip sin c (§ 121), while that of P is Im. Now let F 
denote the volume generated by a unit's area of the plane of the 
figure N moving to a unit's distance along the axis AX, which is 
perpendicular to it, and G the volume described by a unit's area 
of the plane of P moving to a unit's distance along the axis AZ', 
inclined to it at the angle t. Then from the two preceding §§ we 
have — 

: 1, 
Imp, 





Volume VI : F : : m. Ip sin i 


and 


Volume VI : G : : p. Im : i, 


whence 


G : F : : Imp sin t : 


or 


G : F : : sin £ : 1. 



THE MENSUEATION OF VOLUMES. 89 

So, if H were to represent the volume generated by a unit of 
area mo^dng to a unit's distance along a directrix inclined to its 
plane at an angle i\ it might be shown in the same way that — 

H : F : : sin £' : 1, 

whence G : H : : sin £ : sin f' ; 

that is, the volumes described by equal areas, moving to equal dis- 
tances, are as the sines of the inclinations of the directrices to the 
planes of the generating figures. 

174. When the directrix is perpendicular to the plane of the 
generating figure, the sine of its inclination is unity ; hence — 

In the measurement of volume the unit assumed is the 
volume generated by a plane figure of a unit's area, moving 
to a unit's distance along a line perpendicular to its plane. 

175. The content of any volume is its ratio to the unit of the 
volume, or the number of times it contains that unit. Hence — 

The content, or numerical measure of any volume is the 
continued product of the mean area of the generating figure, 
the distance on the directrix over which the latter moves, and 
the sine of the inclination of the directrix to the plane of the 
generating figure. 

176. This formula will now be applied to the simplest case of 
volumes generated by variable areas, — that, namely, in which, in 
the motion of the generating figure, each point of its periphery 
describes a right line, and all these right lines converge in a single 
point, called the apex of the volume. 

In this case, an}^ one of these converging lines may be taken as 
the directrix, or the directrix may be a perpendicular from the 
apex upon the plane of the generating figure ^ Such a perpendicu- 
lar will be in one plane with any one of the lines converging at the 
apex (§ 154). 

177. If a line be drawn from any point of the generating figure 
to the foot of the perpendicular, then in any two positions of the 
generating figure this line will have the same direction. (Thus, in 
Fig. 48, MN is parallel to mil.) For such a line must remain in 
the plane containing the perpendicular AM and the line described 
by the point N, and is also in each position perpendicular to AM 



90 THE MENSURATION OF VOLUMES, 



(§§ 161, 169). Hence the distance (MN or mil) of the point 
from the foot of the perpendicular retains a constant ratio to the 
distance (AM or Am) on the perpendicular from the apex to the 
generating plane. That is, — 

MN : mm :: AM : AmT (§37.) 

So also if S, S, be any other point of the generating figure — 



MS : ms :: AM : Am, 

Whence MN : MS :: mn : msT 

That is, in the motion of the generating figure, the distances be- 
tween its several points retain constant ratios to each other, or 
any one such position or state of the generating figure is a similar 
figure to any other state (§ 123). 

178. From the proportion — 



MN : mn :: AM : Am, 

it is apparent that the ratio of the homologous distances of any two 
states of the generating figure is the ratio of the perpendicular dis- 
tances of their planes from the apex. Or, since the distances of the 
generating plane from the apex, measured upon the perpendicular, 
will be to the distance on any other convergent, as AN, in a con- 
stant ratio, equal to the cosine of the angle which such a con- 
vergent makes with the perpendicular, it appears that the ratio of 
homologous distances, for any two states of the generating figure, 
will be equal to the ratio of the distances of that figure, in its two 
positions, from the apex, measured on the directrix. The ratio of 
the area will be the square of this (§ 123). That is, — 

The area of the generating figure maintains a constant 
ratio to the square of its distance from the apex, measured 
on the directrix. 

^ 179. To find the mean area of the generating figure, and thence 
the measure of the generated volume, let t denote the inclination 
of the directrix to the plane of the generatrix, and li the distance 
to nN (Fig. 48), upon the directrix over which the generating 
figure passes, and let us suppose that the whole distance AN from 



THE MENSURATION OF VOLUMES. 91 

the apex is to nN as 1 to r, where r is a fraction less than 1. 
Then — 

AN = — , and An = AN — nN = li -^— ^. 

If A denote the area of the generating figure when at the point 
N, its area a at the point n may be found from the proportion — 

a . A . . n — . ^ , 

whence a = A (1 — r^. 

Let us now suppose that the distance li is divided into some 
number n of equal parts, then the distance from N to the m*^ point 

of division, counting toward A, will be , hence the distance of 

the same point from A is — 

- — ~ir-' or — (n — rm). 
r li rn ^ * 

The area of the generating figure at the m^^ point of division 
will then be found from the proportion — 



li^ . , Ii' 

r n' 



area : A :: ■ ^,_, (n — rm)' . —^ , 



A 
whence the area sought = -^ (m — rm)'*. 

If now we put for m the numbers 1, 2, 3, 4, etc., successively, 
we find the area of the generating figure at the successive points 
of division to be : — 

A (n- r)S -^ (n - 2i-)% ^ (n - 3r)% etc. 

The sum of all these areas will be — 

A I (n2 — 2 nr + r^ + (n^— 4 rn+4 rO+(n^ — 6 rn+9 r^) 



+ etc. I ; 



92 THE MENSURATION OF VOLUMES. 

or, collecting the terms so as to bring like powers of n together, 
the sum of the areas is — 

— J (m^ + m' + m' + etc. ) — (2 + 4 + 6 + etc. ) m +( 1+4+9 

+ etc.) r^l 

where the number of terms collected in each parenthesis is M. 

Now, it is evident that the sum of n^ -|- n^ -|- 112 -[- etc. to n 
terms is 11 X m^ or n^ ; and it is shown in Algebra * that the sum 
of the series 2, 4, 6, etc., to 11 terms is M^ + n; also, that the 
sum of the series 1, 4, 9, etc., to la terms is — 

^(2ii' + 3ii2 + ii). 
Hence the total sum of the 11 areas is — 

— 2I ii3-_(M2-fii) rii + l- (2 ii^+ 3 112+ n)r2 I ; 



* The first of the two series (2, 4, 6, etc.) is simply an arithmetical pro- 
gression. If the student has not learned the method of summing the second 
series (the series of squares), he may convince himself by the following method 
that the formula for the sum, as given above, is correct. 

First let him test the formula for a few successive values of n, beginning with 
unity. Thus — 

I (2 ii^ -j- 3 n2 ^ n)^ when 11 = 1, is 1 ; when 11 = 2, is 5 ; when n = 3, is 14. 
But 1 + 4=5 and 1 + 4 +9 = 14, 

hence the formula holds true up to three terms. If now the formula be true 
for the sum of the first m. — 1 squares, it is true for one more, i.e., for the sum 
of the first m squares. Fov, substituting m — 1 for 11 in the formula, it be- 
comes — 

i[2(m-l)3+ 3(m-l)^ + (m-l)], 

which reduces to | (2 m^ — 3 mz _|_ j^y 

Now if to this sum of m — 1 square*^ we add the lu*, viz., m.\ the result is 

i (2 m3 -f 8 m^ +m), 

which is evidently what would have been obtained by putting m for n in the 
formula. 

This proves that if the formula correctly represents the sum of any definite 
number of terms, it is true for one more term. If, therefore, it is true for three 
terms, it must be true for four ; if for four, for five, etc. ; whence it is plain 
that it must be true for any number of terms, however great. 

The same method of proof may be applied to show that n^ + n is the sum 
of the progression. 



THE MENSURATION OF VOLUMES. 93 



which, by collecting terms again, may be written in the form — 

The mean of these n areas will now be found by dividing their 
sum by n, and hence is — 

^\(_l-r+ir') n'-(r-ir') n^ + ir'ii], ov 

This formula expresses the mean of the n areas for any value of 
n, great or small ; but if we suppose that some one of the points 
of division falls at every point of the distance li, the value of n 

becomes infinitely great, and the co-efficients - and - — ^ are each 

«qual to zero ; so that all the terms of the formula after the first 
parenthesis vanish, and the true mean value of the variable area 
is found to be — 

A(l-r + ^rO. 
Hence, by § 175, the volume generated is equal to — 
A(l — r+ir^lisinc. 

180. This result is usually expressed in a slightly modified form, 
obtained as follows : If H denote the perpendicular distance be- 
tween the two positions of the generating figure at M and m, and 
8 denote the angle between this perpendicular AM and the direc- 
trix AN, then li cos <5 = H. But since d is the complement of t 
(§ 164) cos ^= sin f, whence li sin : = H. Also, the expression 
A (1 — r + i 1*0 is equivalent to ^ (3 A — 3 Ar + Ar*), which 
may be written — 

HA + A)(l-r) + A(l-2r + rO. 
Now A(l — 2r + rO = A(l — r)' = a, (§179) 

and A (1 — r) = A,/Aa,, 

the mean proportional between A and a. Hence the expression 
for the volume may be written : 

iHCA + VAa + a). 



94 THE MENSURATION OF VOLUMES. 

In words : — 

When a volume is generated by a plane figure moving sa 
that every point of the figure describes a right line, and all 
these right lines converge at one point, then the volume 
generated between any two positions of the generating 
figure, both of which are on the same side of the point of 
convergence, is equal to the product of the perpendicular 
distance between these positions, by one-third the sum of 
the areas of the generating figure in these two positions, 
and of a mean proportional between them. 

181. A line joining two points of the generating figure retains 
the same direction in all positions of the figure, as may easily be 
shown from § 177; and hence in the motion of the generating 
figure such a line describes a plane (§ 12). Hence, when the 
generating figure is bounded by right lines, the surface of the 
generated volume is composed of plane figures (polygons), whose 
area may be found in any given case by the application of the f or- 
mulse for area already deduced (§§ 127, 128). 

182. Besides the species of volume already considered in §§ 
176-181, another variety is generated by a parallelogram moving 
along a directrix which passes through the intersection of its 
diagonals, and is inclined to its plane at an angle c. The sides of 
the parallelogram retain their direction as they move, but vary in 
length, the difference between the lengths of a given side in two 
positions of the generating figure being in a fixed ratio to the dis- 
tance on the directrix intervening between the two positions- 
From these conditions it may be shown that the four vertices of 
the generating figure move in right lines which intersect each other 
two and two, hence the surface of the volume consists of plane 
figures. It may be assumed, with sufficient generality for practi- 
cal purposes, that no two of these lines intersect within the space 
passed over by the generating plane. To find the mean area of 
the generating parallelogram, let p and q be the lengths of two 
adjacent sides at the beginning of the motion, and let p (1 -}- r), 
and q ( 1 -|- r' ) be their length respectively at the end of the 
motion. Then if X denote the angle which they make with each 



THE MENSUKATION OF VOLUMES. 95^ 

other, pq sin X is the area of the parallelogram when the motion 
begins, and pq sin A (1 +r) (1 +i'') is its final area. If the 
distance h. on the directrix which is traversed by the motion be 
divided into n equal parts, then when the generating figure has 

reached the m''' point of division, its sides will be p ( l-\ r ) and 

q (l + ^i*'). and its areapq sin /I (l+^r^ (^+^^')' 

Putting for mthe numbers 1, 2, 3, etc., successively, and expand-^ 
ing the parentheses, we find for the successive values of the area 
the following : — 

pqsinA[l+ 4(i- + r')+^. rr']; 
pq sin A [ 1 + J (r + r') + ^ rr'] ; 

pq sin A [ 1 X ^ (^ + '^') + ~. rr'] ; etc. 

The sum of these n areas will be — 

pqsin / [(1 + 1 + 1 + etc.) + (l + 2 + 3 + etc.) ?i^' 

+(l + 4 + 9 + etc.)^], 

where each parenthesis contains n terms. The sum of n terms of 

the series 1 -j- 1 + 1 + etc. is n, of the series 1 + 2 + 3 -f- etc. is 

»'+n ... . . . . , r. . . . 2ii^+3n^ + ii 

— - — , and of the series 1 + 4 + ^ + etc. is ; 

hence the sum of the n areas is — 

pq sin A [ii+ i^ (n^ + u) (r + r') + g^ (2m3 + 3ii^ 

+11) rr']. 
Expanding the parentheses, arranging the terms according to 
the powers of M, and dividing by n, we obtain for the mean of the 
n areas — 

pq i sin A |(6 + 3 [r + r*] + 2 rrO + i^^±^!±^') 
When the points of division coincide with all the points of the 



96 THE MEI^SURATION OF VOLUMES. 

distance Ii, the number n of these points becomes infinite, and 
the mean area of the parallelogram is found to be — 

i pq sin A (6 + 3 [r + r'] + 2 rv') ; 
whence the volume is — 

i pq sin A (6 + 3 [r + r'] + 2 rr') li sin£, 

or, putting H, the perpendicular distance between the first and 
last positions of the generating plane, in place of la sin «, 

Volume = i H pq sin /I (6 + 3 [r + r'] + 2 rr'). 

The quantity within the parenthesis may be analysed as follows : 

If, in the formula, area = pq sin A (1 + — r) (1 -| r'), 

obtained near the beginning of this §, we put ^ for — , and ex- 
pand the parenthesis, we shall have for the area of the parallelo- 
gram midway between its first and last positions — 

pq sin A (1 4- 1 [r + r'] + i rr'). 

Four times this area is — 

pq sin A (4 + 2 [r + r'] + rr'). 

Also we have for its area in its first position pq sin A, and in its 
last position pq sin A (1 + [r + r] + rr). 

Now, if A denote the area of the moving figure in the first posi- 
tion, a the area in the last position, and M the area when midway 
between the two, it will be found on adding together the above 
values that — 

A + 4 M + a = pq sin A (6 + 3 [r + r'] + 2 rr'). 

Hence the formula for the volume may be written- — 

iH(A + 4M+a). 

183. The formula at the end of § 179 can be analysed in the 
same way. In that §, the area of the generating figure at the m*^ 

point of division was lound to be — ^(m — r iii)^ If we put — 



THE MENSURATION OF VOLUMES. 97 

for m, we find for the area of the figure midway between its ex- 
treme positions — 

whence 4 M z= A ( 4 — 4 r + !•')• 

Also a = A(l — 2r+r0, 

whence A + 4M + a = A(6 — 6r + 2r'), 

and the formula for the volume may be written — 
^H(A + 4M+a). 

184. A volume belonging to either of the two species thus far 
discussed is called in general a frustum. A frustum generated in 
the manner described in § 176 is called si pyramidal frustum, if the 
generating figure be bounded by right lines, but a conical frustum 
when the boundary is a curve. A frustum generated in the man- 
ner described in § 182 is called a, prismoid. The two extreme 
positions of the generating figure, between which the volume is 
contained, are called the bases of the frustum. Any other position 
of the generating figure is a principal section. The perpendicular 
distance between the two bases is the altitude. The formula of 
§§182, 183 afford the following general rule for the mensuration 
of any kind of frustum : — 

The numerical measure of the frustum is found by multi- 
plying one-sixth of the altitude into the sum of the areas of 
the two bases, and of four times the principal section mid- 
way between them. 

185. When two of the lines described by the vertices of the 
parallelogram which generates a prismoid (§ 182) intersect in the 
plane of one- of the bases, that side of the base which is included 
between these vertices vanishes, and the base itself, instead of a 
parallelogram, becomes a portion of a right line, whose area of 
course is zero. The prismoid is in this case called a wedge., and 
the rectilinear distance which represents one of the bases is the 
edge. The length of the remaining base is the length of either of 
its two sides which are parallel to the edge, and the perpendicular 
distance between those two sides is the breadth of the base. In 

7 



98 THE MENSURATION OF VOLUMES. 

the notation of § 182, if p represent the length of the base, then 
then q sine X will be the breadth, the edge will be p (1 + r), and 
the side q ( 1 + r ') will vanish ; that is — 

( 1 -f r') = whence r' = — 1. 

Applying this value of r' to the formula for the volume near the 
end of the § — 

Volume = I H pq sin >i (6 + 3 [r + r'] + 2 rr'), 
the latter becomes — 

i H pq sin /I (6 + 3 [r— 1] —2 r), 
or ^ H pq sin >i (3 + r), 

which may be written — 

iH.qsinA(2p + p[l + r]). 
Whence the rule — 

The measure of the wedge is obtained by adding the edge 
to twice the length of the base, and multiplying the sum 
into one-sixth the product of the breadth of the base and 
the altitude. 

186. The wedge is plainly to be regarded as a limiting case of 
the prismoid, characterized by the fact that the area of one of the 
bases disappears. The other species of frustum has its correspond- 
ing limiting case when the apex is situated at the extremity of the 
directrix. One of the bases then becomes a pointy and its area a 
is 0. In this case the pyramidal frustum becomes a pyrmnid^ and 
the conical frustum a cone^ though the latter name is also applied 
to the surface described by the periphery of the generating figure. 
The measure of the volume is most readily found by the formula 
of § 180, when, as a = 0, ^Asi> = 0, and the formula becomes 
■J All sin J, or ^ All. 

The content of a pyramid or cone is one-third the product 
of the base and altitude. 

187. Another hmiting case of the frustum occurs when r is put 
equal to in the formula of § 179, or when both r = and r'= 
in § 182. The effect is, in either case, to make all the principal 



THE MENSURATION OF VOLUMES. 99 

sections equal to each other. Thus, in § 179, the formula for the 

area of a principal section is — (n — riii)% which becomes A 

when r = 0, and in § 182 the area of a principal section is — 

p<,sin;,(l + ^r)(l+^r'); . 

and this, when r = and r' = 0, becomes pq sin X or A. In this 
case the conical frustum becomes a cylinder (another name used 
for the surface as well as the volume), and a frustum generated by 
a polygon becomes a prism. When the generating polygon is a 
parallelogram (as in the prismoid), a variet}^ of the prism, called 
a parallelopiped, is produced. A parallelopiped whose bases are 
square, whose directrix is perpendicular to the plane of the base, 
and whose altitude is equal to the side of the base, is called a 
cube. 

As all the principal sections, and the two bases are equal, the 
content is readily found from either § 180, § 182, or § 183, for in 
all these formulae, when a = M = A, the expression for the con- 
tent becomes — 

All sin t or AH. 

And this again agrees with the formula of § 175, for the generat- 
ing figure has in this case a constant area. 

The content of any prism or cylinder is the product of 
its base by its altitude. 

The content of the cube is the third power of the side of its base, whence 
the use of the name "cube" as a synonyme for "third power." 

188. A conical frustum generated by the motion of a circle 
along a directrix which passes through its centre and is perpen- 
dicular to its plane, is a right circular conical frustum; and in the 
two limiting cases just noticed becomes a right circular cone, or a 
right circular cylinder. These three volumes are of special im- 
portance as belonging to the class of volumes of revolution, — so 
called because they may be generated not only in the manner 
already described, when the generatrix is a right line, but also 
upon a circular directrix. Thus, in Fig. 50, it is apparent that the 
frustum may be generated not only by the motion of the circle 



100 THE MENSURATION OF VOLUMES. 

along the directrix Oo, but by the revolution of the trapezoid T 
around its side Oo as an axis; in which case every point of the 
revolving figure describes a circle, and one of these circles may be 
taken as the directrix. 

189. In all volumes of revolution, the generating figure is a 
plane figure of constant area and form. In the right circular 
conical frustum, this figure is a trapezoid, whose parallel sides 
are perpendicular to a third side, the axis of revolution. In the 
limiting cases, this trapezoid becomes a right-angled triangle for 
the cone, or a rectangle for the cylinder. That side of the gener- 
ating figure which is not perpendicular to the axis is called the side 
of the frustum, cone, or cylinder. In revolving, the side describes 
a conical or cylindrical surface. The remainder of the surface which 
bounds the volume consists of plane figures (§ 167), — the circular 
bases of the volume. 

190. If the radii of the bases of the frustum be R and r, then 
in the formula of § 180 — 

A=7rR2 and a = tt r^ (§ 145), 

whence \/Aa = - Rr. 

Hence the content of a right circular frustum is — 

Volume = ^ TT H (R^ + Rr + r^). 

When this frustum becomes a cone r ^ 0, and when it becomes 
a cylinder r = R ; so that — 

The content of a right circular cone is -| tt R^jj . ^^^^^ ^f g^ 
right circular cjdinder, tc R^H. 

191. Expressions in terms of H, r, and R may also be obtained 
for the conical or cylindrical surfaces which bound these volumes ; 
and the process will be quite similar, whichever mode of genera- 
tion be adopted. If we regard the circle as the generatrix, and 
take for the directrix a right line drawn from any point of the 
curve to the apex, we must find the mean length of the moving 
periphery, multiply it by the distance over which it moves, and 
this product by the sine of the angle which the curved generatrix 
makes with the rectilinear directrix. Again, if the curve be taken 
as the directrix, that which was formerly the directrix becomes the 



THE MENSURATION OF VOLUMES. 101 

generatrix, and the length of the directrix is different for different 
points of the generatrix. It is therefore necessary to find the 
mean length of the directrix, multiply it by the length of the gen- 
eratrix, and this product by the sine of the angle between them. 
So that the three quantities whose product is to be taken are the 
same in either case, — viz: the side of the frustum, the mean cir- 
cumference of all principal sections, and the sine of the angle 
which the circumference of such a section makes with the side. 

192. First. — To find the side of the frustum. If the radii of 
the bases are R and r, the altitude H, and the side S, the student 
will readily show that — 

In the cone — 

r = .-. S = VH^ + R'. 
In the cylinder — 

r = R.-. S=:H. 

193. Second. — To find the mean circumference of principal sec- 
tions. Any such circumference is the product of its radius by 2 tt 
(§ 145), and its radius is the perpendicular distance of a point on 
the side of the frustum from its axis. Hence the mean distance 
of all such points is to be found, and multiplied by 2 -. If for 
this purpose a number of points be taken on the side, and perpen- 
diculars to the axis be drawn through them, the axis will also be 
divided into equal parts (§ 37). Hence the mean distance of the 
points of the side from the axis is the same as the mean ordinate 
of the side (§ 115), which is the ordinate of its middle point, or 
the mean of the ordinates of its extremities; that is, ^ (R-|-r). 
Hence the mean circumference sought is — 

2 TT X i ( R + r) or tt (R + r). 

In the case of the cone this becomes tt R ; in that of the cylinder, 
2rR. 

[It is to be noted that tlie mean distance of the points of a curve from an axis 
is not in general the same as the mean ordinate of the curve, for the perpen- 
diculars in the former case are drawn so as to cut off equal lengths on the curve, 
and in the latter case on the axis. But ^ 37 shows that for the 7'ight line the 
mean distance and mean ordinate are the same.] 



102 THE MENSURATION OF VOLUMES. 

194. Third. — To find the sine of the angle between the side of 
the frustum and the circumference of a principal section. (This 
angle is not the same as the inclination of the side to the plane of 
the section ; for, as was seen in § 157, it is possible, in a plane 
having a given inclination to a given line, to draw right lines 
making very different angles with it, and the same is true of 
curves.) Let the circumference be divided into n equal parts, 
and let lines (AM, AN, Fig. 50) extend from the apex to two 
successive points of division. These two points are at equal dis- 
tances from the apex, hence the angles, which the two lines make 
with the line MN which joins their extremities, are equal (§ 41, 
Ex. 15) ; and tlie sum of these two angles, together with the angle 
/^, which the two lines make with each other, is equal to 180°. 
Hence the line M N makes with the side of the frustum an angle 

of r . The greater the value of n, the more nearly will M N 

coincide with the arc of the circle, but when n is infinitely great n, 

is zero, hence the angle which the circumference makes with the 

180° 
side of the frustum is — ^— ' or 90°. As the sine of this angle is 1, 

to multiply by it will not affect the product of the other two fac- 
tors, hence the area sought is the product of the side of the frus- 
tum into the mean circumference of principal sections. 

195. The formula for the area of the conical surface of the 
frustum is therefore — 



7rS(R+r) or tt (E + r) V^ + (R — r)'- 
In the case of the cone, since r = 0, the above formula becomes — 

ttES or TiRx/H'-fR'; 

and in the case of the cylinder, where — 

r = R cylindrical surface = 2 :r R H. 

In general, whenever one of two right lines which are in 
the same plane remains fixed in position, and the other re- 
volves about it, so that every point of the revolving line 
describes a circle, the area of the surface generated by 
any distance on the revolving line is the product of that 



THE MENSURATION OF VOLUMES. 103 

distance into the circumference described by its middle 
points. 

The student may show that this rule applies when the lines are perpendicu- 
lar, and will notice that a plane (§ 167) is a variety of conical surface. 

196. A sphere is a volume generated by the revolution of 
a semi-circle about its diameter as an axis. 

The distance of all the points of the surface of a sphere from 
the middle point of this axis (called the centre of the sphere), are 
equal ; and any one of these equal distances is named a radius of 
the sphere. That part of the spherical surface described by any 
arc of the generating circle is a zone^ and the volume generated 
by a figure bounded by an arc and two radii is called a spherical 
sector, 

197. The surface and contents of the sphere may be computed 
by first measuring the area and volume produced by the revolu- 
tion of a triangle whose sides are a chord of the generating figure 
and radii drawn to its extremities. Let the semi-circular curve be 
divided into n equal parts., and let C (in Fig. 51) be the m"' point 
of division, while B is the (m — 1)*'' point. Then if O be the centre, 
and A O H the axis, about which the whole figure revolves, we have 
:first to find the surface S, which will be generated by the revolution 
of the distance BC, and the volume v generated by the triangle 
bounded by BC with OB and OC. Let the angle made with the 
axis by OC be called 2 a, and that made by OB be 2 jS, then, by 
§ 144 — 

2 a = —180'' and 2 /? = 5?^^ 180°. (1) 

Let r denote the radius OA, OB, etc. The line OC makes with 
OB an angle of 2 a — 2 i3. If a line OD be drawn from O equally 
dividing this angle, or making with OB the angle a — j3, its angle 
with OA will be 2i3 + (a — i3), or a + i3. Such a line (§ 141, 
Ex. 16) will meet BC perpendicularly at its middle point D. 
From the points B, C, D, let lines pass perpendicular to the axis ; 
and let the expression, ''the volume under a given distance," be 
understood to mean the volume generated by the revolution of a 
figure bounded by the axis, the given distance, and perpendiculars 



104 THE MENSUKATION OF VOLUMES. 

to the former joining it with the extremities of the latter. The fol- 
lowing are the values of various distances on the figure, which the 
student will verify: — 

BC =2BD-=2rsin(a — 18) and OD = r cos (a — |Q) ; 

KD = OD sin (a + [3) — r cos (a — (3) sin (a + (3) ; 
FB = r sin 2 i3 and OF = r cos 2 (3 ; 

GC = r sin 2 a and OG = r cos 2 a ; 

GF = V (cos 2 i3 — cos 2 a). 

By the rule of § 195, the surface described by BC is equal to 
2 ttKD BC; whence — 

S = 4 77 r' cos (a — )8) sin (a -{- (3) sin (a — (3). (2) 

The factor sin (« + /3) sin (a — (3) may be simplified by first 
expanding, by § 101, to the form — 

(sin a cos (3 + cos a sin [3) (sin a cos (3 — cos a sin j8) ; 
or sin- a cos^ (3 — cos^ a sin^ (3 ; 

which expression, when 1 — sin^ a is put for cos'^ a, and 1 — sin2j3 
for cos^ /3,reduces to the form sin'^ a — sin^ (3. 

sin (a + i3) sin (a — /3) = sin^ a — sin2j3. (3) 

Whence, substituting in (2), 

S = 4-r' cos (a— iS) (silica — sin' (3). (4) 

The volume v may be found by adding together the volumes 
under BC and OC and subtracting that under OB. For these we 
have by § 190 — 

Volume under BC = ^ tt GF . (~GC- + GC . FB + FB^- 

= ^71 r'^ (cos 2 /3 — cos 2 a) (sin' 2 a + sin 2 a sin 2 ^ + sin' 2 (3) 

= ^ TT r^ cos 2 /3 (sin' 2 a + sin 2 a sin 2 (3 -\- sin' 2/3) — i ri v^ cos 

2 a (sin' 2 a + sin 2 a sin 2 [3 + sin' 2 (3) . 

Volume under OC = i - OG . GO = ^ tt r' cos 2 a sin^ 2 a. 

Volume under OB = ^ tt OF . FB^ z= i tt r^ cos 2 /3 sin' 2 f3. 

.-. v=::i"r^cos2i3(sin'2a4-sin 2asin2/3) — ^ ;r r' cos 2 a 

(sin 2 a sin 2 iS + sin^ 2 f3). 



THE MENSURATION OF VOLUMES. 105 

= ^ TT r' sin 2 a COS 2 j3 (sin 2 a + sin 2 /3) — ^ tt r^ cos 2 a sin 2 /J 

(sin 2 a + sin 2 j8). 
zzz^TTV^ (sin 2 a + sin 2 j3) (sin 2 a cos 2 i3 — cos 2 a sin 2 (3), 
Wherefore, from § 110 — 

T=:i^T' (sin 2 « + sin 2)3) sin (2 a — 2/3). (5) 

Now it was found in § 134, equation (1), that when (3 and y are 
any two angles — 

sin i3 + sin ^ = 2 sin i (j3 + y) cos ^ (/3 — y). . 

If 2 a be put for the j8 of this formula, and 2 (3 for y, the formula 
becomes — 

sin2a + sin2i3 = 2 sin (a + jS) cos (« — jS). (6) 

Moreover, by § 102 — sin 2 a =: 2 sin a cos a, 

whence — 
sin (2a — 2i3) = sin 2(a — (3) = 2 sin (« — iS) cos (a— /J). (7) 

Substituting from (6) and (7) in (5), we have — 
v = ^nr^ . 2sin(a + iS)cos(a — 13) . 2 sin (a— /3) cos (a — jS), 
or v = t7rr'cos2(a— |3)sin(a + |3) sin (a — (3). (8) 

Substituting in (8) the value of sin (a-f jS) sin (a — (3) derived 
from (3), we have — 

V = |7rr' cos'^(a — 3) (sin'^a — sin'^iS). (9) 

198. The next step is to find the aggregate surface and volume 
generated by a number of contiguous chords or triangles. For 
this purpose we return to the values of a and /? as found from 
equation (1) of the preceding §, viz: — 

az=z~ 180"' and f3 = ^~^ 180% 

2ia 2]i ' 

whence a—^ = -^ 180°. 

2u 

Substituting these values in equations (4) and (9) we have — 

S = 4 TT r' cos J- 180° ( sin^ ^ 180° — sin^ 5?^:zil80°^ ; 
2ii \ 2ii 2ii / ' 

and V = I TT r^ cos^ J- 180° (^sin^ ^ 180° — sin=^ ^— 180° Y 
2ii\2ii 2ii / 



106 THE MENSURATION OF VOLUMES. 

Replacing in by the numbers 1, 2, 3, 4, etc., successively, we 
have: 
When — 

m = 1, S =- 4 ;r r' cos ^— 180° (^sin' J^ 180**— sin' 0°\ 
' 2ii \ 2ii /' 

when — 

m = 2, s =. 4 r r* cos ^ 180° Ain'* ^ 180° — sin'' J- 180° Y . 
2ii \ 2ii 2n / 

when — 

m = 3, s =- 4 TT r' cos ^ 180° /sin'' ^— 180° — sin'' ^ 180°^ , 

etc., etc. 

It is obvious that when such a series of quantities is added, the 

common factor 4 ;r r^ cos ^ — 180° will be multiplied in the result 

by a parenthesis consisting of only two terms, all the other (inter- 
mediate) terms being cancelled in the addition, so that the sum 
of the surfaces described by p successive chords, the first of which 
is adjacent to the axis, is — 

4 TT r" cos ^ 180° (^sin^ J^ 180° — sin' 0°^ ; 
2ii \ 2ii / ' 

or, if d be put for the angle — 180° which the radius extending to 

the p* point of division makes with the axis, we shall have, — re- 
membering that sin'^^ =^ (1 — cos <5), (§ 104), and that sin 0° 
= 0, (§ 105): — 

sum of p terms = 2 tt r' cos — 180° (1 — cos d). (10) 

Similarly, if y denote the angle - 180° made with the axis by a 

radius extending to the q*'' point of division, the sum of the sur- 
faces described by q successive chords, the first of which is adja- 
cent to the axis, is — 

sum of q terms = 2 tt r' cos ,r— 180° (1 — cos r)- (H) 

Subtracting (11) from (10), we have for the aggregate surface 



THE MENSURATION OF VOLUMES. 107 

•described by the chords extending from the e^^ to the p* point of 

division — ^ 

• 1 

swmo/(p — q) terms 2 t: v^ cos - — 180 (cos y — cos d). (12) 

If in (10) we let p = n; that is, if the successive chords be 
extended along the entire curve of the semi-circle, then the angle 

d or — 180°, becomes 180'^, and cos d becomes — 1 (§ 105) ; whence 

the entire surface described by all the chords of the semi-perime- 
ter is — 

surface = 4 ;r r2 cos ^— 180^ (13) 

Proceeding with the formula for v in Equation (9) in precisely 
the same manner as above with s, we shall find that the aggregate 
volume described by the triangles whose vertices lie between the 
q"* and p"" point of division is — 

5wmo/(p — ii) terms =%7tv^ cos - — 180"" (cos;' — cos ^), (14) 

And that the total volume generated by the revolution of the semi- 
polygon is — 

volume = 1 7r r^ cos — 180**. (15) 

199. If now the series of chords be brought into coincidence 

with the curve by making their number n infinite, then cos - — 180° 

becomes cos 0° or 1. On substituting this value in equations (13) 
and (15) of § 198, the formulae for the surface and content of the 
sphere are obtained. They are — 

surface of sphere = 4 tt r^, 
content of sphere = | tt r'. 
By comparison with the formula of § 145 these results may be 
stated in words as follows : — 

The surface of the sphere is four times the area of a circle 
of equal radius ; and — 

The content of the sphere is the product of its surface by 
one-third its radius. 



108 THE MENSUKATION OF VOLUMES. 

200. The same substitution, of 0° for ^ — 180° reduces equa- 

^ ift ^ 

tions (12) and (14) of § 198 to formulae for the area of a zone and 

the content of a spherical sector. These formulae are — 

area of a zone = 2 - r^ (cos y — cos o) ; and 

splierical sector = f tt r^ (cos y — cos d). 

The formula for the zone may be factored, and written thus — 

2 rr r (r cos y — r cos (5). 

In Fig. 51, if the angle of OB with the axis be called ^, and 

that of OC be ^, then the formula represents the area of the zone 

described by the revolution of the arc BC. The factor 2 tt r is the 

periphery of a circle whose radius is i* (§ 145), while the factor 

(r cos y — V cos d) represents the distance OF — ■ OG or GF, the 

distance cut off upon the axis between perpendiculars from the 

extremities of the generating arc BC. This distance is called the 

altitude of the zone. Hence — 

The area of a zone is the product of its altitude by the 
periphery of a circle whose radius is that of the sphere ; 
and — 

The content of a spherical sector is the product of the 
area of its zone by one-third the radius of the sphere. 

Examples. 

1. The p3Tamid of Cheops was built on a square base, each of 
the sides of which was 764 feet long. The summit, vertically 
over the centre of the base, had originally a height of 479 feet. 
Required the area of the base and of each face, and the volume of 
masonry composing the structure. 

2. A cubic foot of water weighs a thousand ounces ; required 
how many pounds are contained in a tank six feet long and four 
feet broad, when it is filled to a depth of three feet. 

3. Required the surface and weight of an upright hexagonal 
block of Quincy granite (specific gravity 2.652) each side of 
which measures 18 inches and the height is 3 feet. 

4. What is the weight of a cylindrical column of the same ma- 
terial, 4 feet long and 15 inclies in diameter? 



THE MENSURATION OF VOLUMES. 109 

5. A certain cask is composed of two equal conical frustums 
joined at their larger bases. The largest diameter is 28 inches, 
the diameter of the head 20 inches, and the length 40 inches ; how 
many gallons of wine will it hold, there being 231 cubic inches in 
a gallon ? 

6. What must be the diameter of a cylindrical quart cup, 6 
inches deep? 

7. How many cubic feet of earth are contained in an embank- 
ment 10 feet high, having a level top 20 feet wide and 40 feet long, 
and resting on a level base 40 feet wide and 60 feet long? 

8. Find the surface and content of a cone whose base is a circle 
having a circumference of 1 foot 10 inches, and whose altitude is 
12 inches. 

9. At the mean distance of the earth from the sun (which is 
supposed to be about 93,000,000 miles), his apparent semi-diame- 
ter is 16' 01".5. Required the diameter and circumference of the 
sun in miles, his volume, and the area of his surface. 

10. On a school-globe, 1 foot in diameter, the tropics are drawn 
at a latitude of 23^° and the polar circles at a latitude of 66^°. 
What is the area of each zone bounded by these circles ? 

11. Within a right circular cylinder (Fig. 52), whose altitude 
is equal to the diameter of its base, is inscribed a sphere which 
touches the two bases and the cylindrical surface. What is the 
ratio of the surface of the sphere to the entire surface of the 
cylinder? What is the ratio of their contents? 

[The student will find the two ratios equal, each being 2:3. 
This relation between the two volumes is said to have been dis- 
covered by Archimedes, who was so much interested in the investi- 
gation that he directed it to be commemorated in the device upon 
his gravestone. Cicero, in his Tusculan Disputations, relates that 
by this mark he discovered the neglected grave of the mathemati- 
cian, more than a century after his death.] 



\D S GEOMETRY 



PLATE I 




Fig. 22 



^-c 



§ 72-82 & 85 



E 


Y 


Fig. 23 




/^-""'^ 


A 




For § 9G 



R X 



LOUD S GEO]>IETRY 



PLATE I 




I 


N 


t-lg 


Vi 


r 




A 


B/ 




^ 


M 


) 


X 




For k 


r 1 







For g 97 



DS GEOMETRY 



Fig. 28 



Fig. 29 




LOUDS GEOMETRY 



PLATE II 
























0' 




^ o^ 

,;%'-^</:';,-:,%:-'->°:. 



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